# Wishartness of quadratic forms: a characterization via Jordan algebra representations.

1. Introduction

This paper is a sequel to Masaro and Wong (12) and, like (12), is based on Wong and Masaro (22).

Consider a normally distributed random matrix Y with mean 0 and general covariance matrix [[SIGMA].sub.Y]. This paper deals with the following question.

(i) When does a quadratic form in Y, Q(Y), follow a Wishart distribution with m degrees of freedom and scale parameter [SIGMA]?

In the case where Y is a real random matrix and Q(Y) = Y'WY with W nonnegative definite question (i) has been addressed by various authors; see for example, Khatri (8), DeGunst (3), Pavur (17), Wong and Wang (21) and Mathew and Nordstrom (16). For the most general case, i.e. W is symmetric and [[SIGMA].sub.Y] and [SIGMA] are nonnegative definite, Wong, Masaro and Wang (20) obtained some results by placing restrictions on the image space of [[SIGMA].sub.Y]. These restrictions were lifted in Masaro and Wong (11) but the necessary and sufficient conditions obtained for the Wishartness of Y'WY were quite complicated. It was then discovered (Wong and Masaro (22) and Masaro and Wong (12)) that these conditions reflected the fact that the Wishartness of Y'WY depended on whether or not a certain linear transformation was a Jordan algebra homomorphism.

We shall exploit the connection between Jordan algebras and the Wishart distribution, and characterize the Wishartness of a quadratic form Q(Y) in terms of Jordan algebra homomorphisms or more precisely, Jordan algebra representations. The framework of Jordan algebras provides a unified approach to question (i) in that the cases where Y is a real, complex or quaternionic normal random matrix can be dealt with simultaneously as part of a general theory. In addition we feel that this point of view provides a deeper insight into the nature of the Wishart distribution.

The outline of this paper is as follows. In Section 2 we set up some notation and recall the definitions of the normal and Wishart distributions. Section 3 summarizes the results from the theory of Jordan algebras that we will need. The key result of this section is Theorem 3.5.2 which links the concept of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution. In Section 4 we give our main results (Theorem 4.4 and Corollary 4.5) which characterize the Wishartness of Q(Y).

We remark that the applications of Jordan algebras to statistics is not new. See Seely (19), Malley (10), Jensen (7), Massam (13), Massam and Neher (14), (15) and Letac and Massam (9).

2. Preliminaries

Let [A.sub.d] denote R, C or H according to d = 1, 2 or 4, where R denotes the field of real numbers, C the field of complex numbers and H the division ring of quaternions over R. Each x [member of] H may be represented as x = [x.sub.1]+i[x.sub.2]+j[x.sub.3]+k[x.sub.4], where [x.sub.i] [member of] R, [i.sup.2] = [j.sup.2] = [k.sup.2] = -1, ij = -ji = k, jk = -kj = i and ki = -ik = j. The conjugate of x is [bar.x] = [x.sub.1]-i[x.sub.2]-j[x.sub.3]-k[x.sub.4] and the real part of x is Re x = [x.sub.1]. Analogous definitions apply to C. Note that multiplication in H is associative but not commutative and that for x, y [Note that multiplication] H, [bar.xy] = [bar.y] [bar.x].

Let [M.sub.[nxp].sup.d] denote the family of n X p matrices over [A.sub.d]. A matrix A [member of] [M.sub.[nxp].sup.4] may be written as A = [A.sub.1] + i[A.sub.2] + j[A.sub.3] + k[A.sub.4], where [A.sub.i] [member of] [M.sub.[nxp].sup.1]. The transpose, conjugate and adjoint of A are defined by A' = [A'.sub.1] + i[A'.sub.2] + j[A'.sub.3] + k[A'.sub.4], [bar.A] = [A.sub.1]-i[A.sub.2]-j[A.sub.3]-k[A.sub.4] and A* = [bar.A'] respectively. (Here [A'.sub.i] is the usual matrix transpose for matrices over R). Similar notions apply to [M.sub.[nxp].sup.d] for d = 1, 2. Of course for A [member of] [M.sub.[nxp].sup.1], [bar.A] = A, A* = A' and Re A = A.

We remark that the familiar formulas (AB)' = B'A' and [bar.AB] = [bar.A] [bar.B] hold only in the cases d = 1 or 2 but the formulas (AB)* = B* A* and Re Tr(AB) = Re Tr(BA) hold for d = 1, 2 or 4.

A matrix A [member of] [M.sub.[nxn].sup.d] is called Hermitian if A* = A. The family of n x n Hermitian matrices over [A.sub.d] will be denoted by [H.sub.n.sup.d].

We shall view [M.sub.[nxp].sup.d] as a Euclidean vector space, i.e., a vector space over R with the inner product (A, B) = ReTr(AB*). Thus the dimension of [M.sub.[nxp].sup.d] is npd. Also we have (A, B) = ([bar.A], [bar.B]) and (A, BC) = (B* A, C) = (AC*, B) whenever the matrix multiplication is defined.

Let End([M.sub.[nxp].sup.d]) denote the (real) vector space of endomorphisms of [M.sub.[nxp].sup.d]. The adjoint of T [member of] End([M.sub.[nxp].sup.d]) is denoted by T*; so for all X, Y [member of] [M.sub.[nxp].sup.d], (T(X), Y) = (X, T*(Y)). T is called self-adjoint of T* = T. The space of self-adjoint endomorphisms will be denoted by [End.sub.S]([M.sub.[nxp].sup.d]). For A [member of] [M.sub.[nxn].sup.d] and B [member of] [M.sub.[pxp].sup.d], the Kronecker product A [cross product] B is defined as the element in End([M.sub.[nxp].sup.d]) such that

(A [cross product] B)(C) = ACB*.

For X [member of] [M.sub.[nxp].sup.d], let [delta](X) (or [X]) be the coordinate vector of X in [R.sup.npd] with respect to an orthogonal basis for [M.sub.[nxp].sup.d] and for T [member of] End([M.sub.[nxp].sup.d]), let [phi](T) (or [T]) be the matrix representation with respect to this basis. Thus (X, Y) = ([delta](X), [delta](Y)) and [delta](T(X)) = [phi](T)[delta](X). Also [phi](T*) = [phi](T)' and T is self-adjoint (nonnegative definite, positive definite) according as [phi](T) is symmetric (nonnegative definite, positive definite).

We shall view End([M.sub.[nxp].sup.d]) as a real inner product space with inner product

(S, T) = ([phi](S), [phi](T)) = Tr([phi](S)[phi](T)').

One may also write (S, T) = Tr(ST*) since by definition, Tr(ST*) = Tr [phi](ST*) = Tr([phi](S)[phi](T)').

The following lemma will be useful.

Lemma 2.1. Let A, C [member of] [M.sub.[nxn].sup.d] and B, D [member of] [M.sub.[pxp].sup.d]. Then:

(a) (A [cross product] B, C [cross product] D) = d(A, C)(B,D).

(b) Tr(A [cross product] B) = d Re Tr(A)Re Tr(B).

Proof. We shall merely prove the result for the case d = 4.

(a) Let B be the standard orthogonal basis for [M.sub.[nxp].sup.d]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Now each basis element in B is of the form F = [alpha][F.sub.rs], where [alpha] = 1, i, j or k and [F.sub.rs] is the n x p matrix with (r, s)th entry equal to 1 and all other entries 0. Thus

F* CA* F = [bar.[alpha]][F.sub.sr]CA* [alpha][F.sub.rs] = [bar.[alpha]][u.sub.rr][alpha][E.sub.ss]

where [u.sub.rr] is the (r, r)th entry of CA* and [E.sub.ss] is the p x p matrix with (s, s)th entry equal to 1 and other entries 0. Then using the fact that [[SIGMA].sub.[alpha]] [bar.[alpha]][u.sub.rr][alpha] = d Re [u.sub.rr] (d = 4 here), we have

[summation over (F [member of] B)]F*CA*F = [summation over (s)][summation over (r)][summation over ([alpha])][bar.[alpha]][u.sub.rr][alpha][E.sub.ss] = d Re Tr(CA*)[I.sub.p].

Thus by (2.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) Since Tr(A [cross product] B) = (A [cross product] B, [I.sub.n] [cross product] [I.sub.p]), part (b) follows by taking C = [I.sub.n] and D = [I.sub.p] in (a).

Let [N.sub.k]([gamma], [SIGMA]) denote the usual normal distribution over [R.sup.k] with mean [gamma] and nonnegative definite covariance matrix [SIGMA]. A random variable Y taking values in [M.sub.[nxp].sup.d], d = 1, 2 or 4, is said to have a real, complex or quaternionic normal distribution with mean [[mu].sub.Y] [member of] [M.sub.[nxp].sup.d] and covariance matrix [[SIGMA].sub.Y] [member of] [End.sub.S]([M.sub.[nxp].sup.d]) if [delta](Y) ~ [N.sub.npd]([delta]([[mu].sub.Y]), 1/d[phi]([[SIGMA].sub.Y])). In this case, we write Y ~ [N.sub.[nxp].sup.d]([mu], [[SIGMA].sub.Y]). Note that [N.sub.k]([mu], [SIGMA]) = [N.sub.[kx1].sup.1]([mu], [SIGMA] [cross product] 1).

A random variable U taking values in [H.sub.p.sup.d], d = 1, 2, or 4, is said to have a real, complex or quaternionic Wishart distribution with m degrees of freedom and scale matrix [SIGMA] [member of] [H.sub.p.sup.d] if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where Z ~ [N.sub.[mxp].sup.d](0, [I.sub.m] [cross product] [SIGMA]). In this case, we write U ~ [W.sub.p.sup.d] (m, [SIGMA]).

Remark 2.1. Usually, in the case d = 2, one defines the [W.sub.p.sup.2] (n, [SIGMA]) distribution to be the distribution of [n.summation over (i=1)] [Y.sub.i][Y.sub.i]*, where the [Y.sub.i]'s are iid [N.sub.[px1].sup.2](0, [SIGMA] [cross product] 1) (see Goodman (5)). Note that one may also write [n.summation over (i=1)] [Y.sub.i][Y.sub.i]* as Y'[bar.Y], where Y' = [[Y.sub.1], [Y.sub.2], ..., [Y.sub.n]]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then Z*Z = Y'[bar.Y] and since the [Y.sub.i]*'s are iid [N.sub.[1xp].sup.2](0, 1 [cross product] [SIGMA]), Z ~ [N.sub.[nxp].sup.2](0, [I.sub.n] [cross product] [SIGMA]). Let [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) be a linear map.

For each y [member of] [M.sub.[nxp].sup.d], the linear form on the real vector space [H.sub.p.sup.d] defined by t [right arrow] (y, [psi](t)y) is given by an inner product on [H.sub.p.sup.d]. (For simplicity, [psi](t)(y) is abbreviated as [psi](t)y.) Thus there is an element in [H.sub.p.sup.d] depending y and [psi], call it [Q.sub.[psi]](y), such that

(y, [psi](t)y) = (t, [Q.sub.[psi]](y)) (2.2)

for all t [member of] [H.sub.p.sup.d]. We call the map [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] the [H.sub.p.sup.d]-valued quadratic form associated with the linear map [psi].

If Y ~ [N.sub.[nxp].sup.d] (0, [[SIGMA].sub.Y]), then [Q.sub.[psi]](Y) is a random quadratic form taking values in [H.sub.p.sup.d]. The mean of [Q.sub.[psi]](Y) can be obtained as follows:

(t, E([Q.sub.[psi]](Y))) = [1/d]([[SIGMA].sub.Y],[psi](t)). (2.3)

Indeed, (t, E([Q.sub.[psi]](Y))) = E(t,[Q.sub.[psi]](Y)) = E(Y, [psi](t)(Y))=E([delta](Y), [phi]([psi](t))[delta](Y)) = E([delta](Y)[delta](Y)', [phi]([psi](t))) = (1/d[phi]([[SIGMA].sub.Y]), [phi]([psi](t))) = 1/d ([[SIGMA].sub.Y]), [psi](t)).

The main purpose of this paper is to determine necessary and sufficient conditions for [Q.sub.[psi]](Y) to have a Wishart distribution. These conditions are closely linked to the natural Jordan algebra structure on [H.sub.p.sup.d].

We shall now give an example of random quadratic form.

Example 2.1.Let Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]), W [member of] [H.sub.p.sup.d] and [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]), [psi](t) = W [cross product] t. Then (t, [Q.sub.[psi]](Y)) = (Y, [psi](t)(Y)) = (Y, WYt*) = (Y*W*Y, t*) = (Y*WY, t).

Thus [Q.sub.[psi]](Y) = Y* WY. Further, in the case W = [I.sub.n] and [[SIGMA].sub.Y] = [I.sub.n] [cross product] [SIGMA], we have [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d] (n, [SIGMA]) and by (2.3) and Lemma 2.1, (t, E([Q.sub.[psi]](Y))) = 1/d([I.sub.n] [cross product] [SIGMA], [I.sub.n] [cross product] t) = 1/d nd([SIGMA], t). Hence

E([Q.sub.[psi]](Y)) = n[SIGMA]. (2.4)

3. Jordan Algebras

In the following subsections, we shall summarize the notions and results from the theory of Jordan algebras that we require. Only some results are proven. Other results are known and can be found in Braun and Koecher (1), Faraut and Koranyi (4) or Jacobson (6).

The key result in this section is Theorem 3.5.2 which connects the notion of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution.

3.1 Basic Definitions

A Jordan algebra V over the set R of real numbers is a real vector space with a product ab such that ab = ba, [lambda](ab) = ([lambda]a)b, ([a.sub.1] +[a.sub.2])b = [a.sub.1]b+[a.sub.2]b and a([a.sup.2]b) = [a.sup.2](ab) for [lambda] in R and a, [a.sub.1], [a.sub.2] and b in V. An element e in V will be called an identity if ex = x for all x in V.

Let R[X] denote the algebra over R of polynomials in one variable with coefficients in R. For an element x in V, we define R[x] = {p(x): p(X) [member of] R[X]}. The product ab is, in general, nonassociative. In V, define [x.sup.n] recursively by [x.sup.n] = x[x.sup.[n-1]]. V, as Jordan algebra, is power associative, that is [x.sup.p+q] = [x.sup.p][x.sup.q] for all x in V and positive integers p, q. It can be shown that for any p(x) and q(x) in R[x], and a [member of] V, p(x)(q(x)a) = q(x)(p(x)a). An element x in V is said to be invertible with inverse y (denoted [x.sup.-1]) if xy = e and y belongs to R[x]. An element c in V is an idempotent if [c.sup.2] = c. Two idempotents c and d are called orthogonal if cd = 0. An idempotent said to be primitive if it is non-zero and cannot be written as the sum of two non-zero idempotents. A set of idempotents {[c.sub.1], [c.sub.2], ..., [c.sub.m]} in V is called complete if [c.sub.i][c.sub.j] = 0 for i [not equal to] j and [[SIGMA].sub.1.sup.m] [c.sub.i] = e. If, in addition, each [c.sub.i] in this set is primitive the set is called a Jordan frame. A mapping [phi] from V to a Jordan algebra (W, #) is a homomorphism if [phi] is linear and [phi](ab) = [phi](a) # [phi](b) for all a, b in V. Since the polarization identity xy = [[(x + y).sup.2] - [(x - y).sup.2]]/4 holds in V, the linear map [phi] will be a homomorphism iff [phi]([a.sup.2]) = [phi](a) # [phi](a) for all a in V. If V possesses an identity e, [phi](e) will be an identity for [I.sub.m] [phi](the image space of [phi]) but [phi] need not be an identity for W. If [phi] is one to one and onto W, then [phi] is called an isomorphism of V onto W and V and W are said to be isomorphic. A subset I of V is an ideal in V if I is a linear subspace of V and for any x in I, y in V, both xy and yx belong to I; V is said to be simple if its only ideals are {0} and V itself.

It can be shown (see Jacobson (6)) that there exists a unique integer

r > 0 and unique functions [a.sub.j]: V [right arrow] R such that the [a.sub.j] are homogeneous of degree j and for all x in V,

[x.sup.r] - [a.sub.1](x)[x.sup.r-1] + [a.sub.2](x)[x.sup.r-2] - ... + [(-1).sup.r][a.sub.r](x) = 0.

The polynomial

[m.sub.x](X) = [X.sup.r] - [a.sub.1](x)[X.sup.r-1] + [a.sub.2](x)[X.sup.r-2] - ... + [(-1).sup.r][a.sub.r](x)

is called the generic minimum polynomial for x; the degree 'r' of [m.sub.x](X) is called the rank of the Jordan algebra V. The generic trace and generic determinant of x in V are defined by

tr(x) = [a.sub.1](x) and det(x) = [a.sub.r](x).

We shall use upper case notation, "Det", "Tr" to denote the usual trace and determinant for matrices (endomorphisms) and lower case notation "det", "tr" to denote the generic trace and determinant of an element in a Jordan algebra. When required, the notation "[tr.sub.W]"and "[det.sub.W]" will be used to denote the generic trace and determinant with respect to a specific Jordan algebra W.

The space [H.sub.r.sup.1] of Hermitian (symmetric) r x r matrices over R is a Jordan algebra when endowed with the Jordan product A [omicron] B = [AB + BA]/2, with the product on the right side being the usual matrix product. In this case, the generic trace and determinant correspond to the usual trace and determinant for matrices. The notion of Jordan algebra is motivated from this special case.

3.2 Euclidean Jordan Algebras

We say that V is Euclidean if there exists an inner product (., .) on V which is associative. That is (ab, c) = (b, ac) for all a, b and c in V. In every Euclidean Jordan algebra with identity, the generic trace form, (x, y) [right arrow] tr(xy) is positive definite and associative. Unless otherwise stated, we assume that the inner product in a finite dimensional Euclidean Jordan algebra V with identity is given by (x, y) = tr(xy), x, y [member of] V.

For x in V the linear map L(x): V [right arrow] V is defined by L(x)(v) = xv. (Again, for simplicity, L(x)(v) may be written as Lxv. We shall not repeat such convention further.) Further, we define P(x) = 2L[(x).sup.2] - L([x.sup.2]) and P(x, y) = L(x)L(y)+L(y)L(x)-L(xy). Since the inner product is associative, L(x), P(x) and P(x, y) are self-adjoint. The map x [right arrow] P(x) is called the quadratic representation of V. In the Jordan algebra [H.sub.r.sup.1] (with product A [omicron] B = [AB + BA]/2), we have

P(A)B = ABA, P(A,B)C = [ACB+BCA]/2 and P(ABA) = P(A)P(B)P(A).

An element x in V is said to be positive definite (nonnegative definite) if L(x) is positive definite (nonnegative definite). The symmetric cone associated with V is the interior of the set Q = {[x.sup.2]: x [member of] V} and is denoted by [OMEGA](V). It is known that [OMEGA](V) = {x [member of] V: x is positive definite} and [bar.[OMEGA]](V) = Q = {x [member of] V: x is nonnegative definite}.

The following results are well known.

1. For any x in V, x is invertible iff P(x) is invertible whence P[(x).sup.-1] = P([x.sup.-1]) (3.1)

2. For any x, y in V, tr(P(x)y) = tr([x.sup.2]y). (3.2)

3. For any x, y, z in V, P(P(z)x, P(z)y) = P(z)P(x, y)P(z). (3.3)

4. For any x in V and p(x) and q(x) in R[x], P(p(x)q(x)) = P(p(x))P(q(x)) (3.4)

3.3. Pierce and Spectral Decompositions

For an idempotent c in V, the Pierce spaces V(c, i) are defined by

V (c, i) = {x [member of] V: cx = ix}, for i = 0, 1/2, 1.

It is well known that

V = V(c, 1) [direct sum] V(c, 1/2) [direct sum] V(c, 0) (a vector space direct sum)

This decomposition is orthogonal with respect to any associative inner product on V. Also, V(c, 1) and V(c, 0) are Jordan subalgebras of V and c is an identity for V(c, 1). The projections in the Pierce decompositions onto V(c, 1), V(c, 1/2) and V(c, 0) are P(c), I-P(c)-P(e-c) and P(e-c) respectively.

For an illustration of these concepts, take V = [H.sub.r.sup.1], r = p + q and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If V is simple, the value d = dim[V(a, 1/2) [intersection] V(b, 1/2)] is invariant for any pair of orthogonal primitive idempotents (a, b). The value d is called the Pierce invariant and it is related to the dimension and rank of V by n = r+r(r-1)d/2. Moreover, when V is simple, so is V(c, 1) and if c is not primitive, the Pierce invariant for V(c, 1) is also equal to d.

Suppose that rank (V) = r. Then for each x in V, there exists a Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} and [[lambda].sub.i] [member of] R such that

x = [[lambda].sub.1][c.sub.1] + [[lambda].sub.2][c.sub.2] + ... + [[lambda].sub.r][c.sub.r]. (3.5)

The numbers [[lambda].sub.i] (with their multiplicities) are uniquely determined by x and are called the eigenvalues of x. Further, tr(x) = [SIGMA][[lambda].sub.i] and det(x) = [PI][[lambda].sub.i]. The decomposition (3.5) is called the spectral decomposition of x. The rank of x, rk(x), is the number of non-zero eigenvalues (with multiplicities counted) in its spectral decomposition.

For any real-valued function f continuous on a closed interval containing {[[lambda].sub.i]: i = 1, 2, ... r}, define

[~.f](x) = [r.summation over (i = 1)]f([[lambda].sub.i])[c.sub.i]. (3.6)

To check that [~.f](x) is well-defined, i.e., is independent of the Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} chosen for x, let p(X) be a polynomial in R[X]. Then it is straitforward to prove that

[~.p](x) = p(x), (3.7)

which is independent of the chosen Jordan frame for x. So by the Weierstrass approximation theorem, [~.f](x) in (3.6) is determined by f and x. For convenience, we shall write f(x) instead of [~.f](x).

Let

[x.sup.+] = [summation over ([[lambda].sub.i] [not equal to] 0)][[lambda].sub.i.sup.-1]][c.sub.i], [x.sup.0] = [summation over ([[lambda].sub.i] [not equal to] 0)][c.sub.i] and [x.sup.[alpha]] = [summation over ([[lambda].sub.i] [not equal to] 0)][[lambda].sub.i.sup.[alpha]][c.sub.i] (3.8)

where [alpha] may be any real number if all [[lambda].sub.i] are positive and [alpha] may be an integer if some [[lambda].sub.i] are negative. Then [x.sup.+], [x.sup.0], [x.sup.[alpha]] are also well-defined since they may be expressed as f(x) where f is continuous in a closed interval containing the eigenvalues of x: suppose that x = [n.summation over (i=1)] [[lambda].sub.i][c.sub.i]. For simplicity say 0 < [[lambda].sub.1] [less than or equal to] [[lambda].sub.2], ... [less than or equal to] [[lambda].sub.k] with all other [[lambda].sub.i] = 0. Let f(t) = [[lambda].sub.1.sup.[[alpha]-1]]t for 0 [less than or equal to] [[lambda].sub.1] and f(t) = [t.sup.[alpha]] for t > [[lambda].sub.1]. Then f is continuous on [0, [[lambda].sub.k]] with f(0) = 0 and so f(x) = [x.sup.[alpha]] is well-defined. The case where some [[lambda].sub.i] are negative and [alpha] is an integer is similar.

Our motivation for (3.8) arises from the notion of Moore-Penrose inverse in linear algebra and the consideration of Wishart distributions with singular scale matrix.

The following lemmas will be needed later on.

Lemma 3.3.1. Let V be a Euclidean Jordan algebra of rank r with identity e. Then for x [member of] V and functions f and g such that f(x) and g(x) are continuous on a closed interval containing the eigenvalues of x, P(f(x))P(g(x)) = P(f(x)g(x)). In particular, for any u in [bar.[OMEGA]](V), P([u.sup.[alpha]])P([u.sup.[beta]]) = P([u.sup.[alpha]+[beta]]), [alpha], [beta] [member of] R.

Proof. This follows from (3.4) and the Weierstrass approximation theorem.

Lemma 3.3.2. Let V be a Euclidean Jordan algebra of rank r with identity e. (a) Let c [epsilon] V be an idempotent. Then for all x [epsilon] V (c, 1),

[tr.sub.c](x) = [tr.sub.v](x) and [det.sub.c](x) = [det.sub.v](e-c + x).

where [tr.sub.c] and [det.sub.c] are the generic trace and determinant for V (c, 1).

(b) Suppose that the set {[c.sub.1], [c.sub.2], ..., [c.sub.s]} is a family of non-zero orthogonal idempotents (not necessarily primitive) in V and x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i], [[lambda].sub.i] [epsilon] R. Then

[det.sub.V](e-x) = [[product].sub.1.sup.s][(1-[[lambda].sub.i]).sup.tr([c.sub.i])].

Proof. (a) Let x [epsilon] V (c, 1) and suppose that x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i] is a spectral decomposition of x in the Jordan algebra V (c, 1). Then c = [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] since c is the identity for V (c, 1). Extend {[c.sub.1], [c.sub.2], ..., [c.sub.s]} to a Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.s], [c.sub.s+1], ..., [c.sub.r]} for V. Then it is clear that [tr.sub.c](x) = [tr.sub.V] (x). Since e - c + x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i] + [[SIGMA].sup.r.sub.[i=1]] [c.sub.i] we have, [det.sub.c](x) = [det.sub.V](e - c + x).

(b) Let [c.sub.s+1] = e - [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] (if [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] = e, s = r and there is nothing to prove). Then {[c.sub.1], [c.sub.2], ..., [c.sub.s+1] is a complete system of orthogonal idempotents. For each i = 1, 2, ..., s+1, let [c.sub.i] = [[SIGMA].sup.ni.sub.[j=1]] [d.sub.ij] be a spectral decomposition on [c.sub.i]. Then [n.sub.i] = tr([c.sub.i]) and the set {[d.sub.ij]: i = 1,2, ..., s+1, j = 1,2, ..., [n.sub.i]} is a Jordan frame for V. Hence

e - x = [[SIGMA].sup.s.sub.[i=1]] [[SIGMA].sup.ni.sub.[j=1]](1-[[lambda].sub.i])[d.sub.ij] + [c.sub.s+1] from which (a) follow.

3.4. Representations, Simple Euclidean Algebras

Let V be a Jordan algebra over R and E a vector space over R. A representation of V on E is a linear map [tau]: V [right arrow] End(E) such that

[tau](xy) = [1/2]([tau](x)[tau](y) + [tau](y)[tau](x)),

i.e. the map [tau] is a Jordan algebra homomorphism of V into End(E) equipped with the Jordan product A [omicron] B = 1/2 (AB + BA). The representation [tau] is said to be self-adjoint if for any x [epsilon] V, [tau] (x) is a self-adjoint endomorphism of E.

It can be proved that there are only four types of simple Euclidean Jordan algebras that admit a self-adjoint representation, namely the spaces [H.sub.r.sup.d], d = 1, 2 and 4 when endowed with the Jordan product A[omicron]B = 1/2 (AB+BA) and the space R x W, where W is a finite-dimensional inner product space, equipped with the product ([[lambda].sub.1], [w.sub.1]) [omicron] ([[lambda].sub.2], [w.sub.2]) = ([[lambda].sub.1][[lambda].sub.2] + ([w.sub.1], [w.sub.2]), [[lambda].sub.1][w.sub.2] + [[lambda].sub.2][w.sub.1]). The Euclidean Jordan algebra [H.sub.3.sup.8] of 3 x 3 Hermitian matrices over the Octonions is simple with Pierce constant d = 8 but does not admit a representation.

If V is a simple Euclidean Jordan algebra of rank r and dimension n, then the following isomorphisms [equivalent] hold.

1. If r = 3, V [equivalent] [H.sub.3.sup.d], d = 1, 2, 4 or 8.

2. If r [greater than or equal to] 4, V [equivalent] [H.sub.r.sup.d], d = 1, 2, or 4.

3. If r = 2, V [equivalent] R x [R.sup.n-1], d = n-2 and in the case that d = 1, 2 or 4(i.e. n = 3,4 or 6), V [equivalent] [H.sub.2.sup.d] [equivalent] R x [R.sup.n-1].

4. If r = 1, V [equivalent] R.

For the Jordan algebras [H.sub.r.sup.d], d = 1, 2, 4, the generic trace and corresponding inner product are given by trA = TrA and (A, B) = tr(A[omicron]B). Note that since A and B are Hermitian, (A, B) = Re Tr(AB) = Re Tr(AB*) so that the inner product for [H.sub.r.sup.d] is simply the inner product inherited from the space [M.sub.[rxr].sup.d] as described in section 2. The generic determinant for [H.sub.r.sup.d] may be obtained from

[(det A).sup.d] = Det [tau](A),

where [tau] is the standard representation for [H.sub.r.sup.d], that is:

1. For A [epsilon] [H.sub.r.sup.1], [tau](A) = A.

2. For A = [A.sub.1] + i[A.sub.2] [epsilon] [H.sub.r.sup.2], [tau] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. For A = [A.sub.1] + i[A.sub.2] + j[A.sub.3] + k[A.sub.4] [epsilon] [H.sub.r.sup.4], [tau] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We need the following lemma.

Lemma 3.4.1. Let V be a simple Euclidean Jordan algebra of rank r [greater than or equal to] 2 with Pierce constant d and identity e. Suppose [tau]: V [right arrow] End(E) is a self-adjoint representation of V on E. Then Tr([tau](e)) is divisible by rd.

Proof. Since [tau](e) is an orthogonal projection, we may choose an orthogonal basis for E such that [tau](e) = diag[[I.sub.k], 0]. Also, since for any x [epsilon] V, [tau](x) = [tau](ex) = 1/2 [[tau](e)[tau](x) + [tau](x)[tau](e)], we must have [tau](x) = diag[[[tau].sub.1](x), 0], where [[tau].sub.1](x) is a k x k matrix. Then [[tau].sub.1] is a self-adjoint representation of V on [R.sup.k] with [[tau].sub.1](e) = [I.sub.k]. The desired result then follows from Theorem 3 and Theorem 6 of Jensen (7).

3.5. Mutations

Let V be a Euclidean Jordan algebra with identity e. Given an element u in V one may define a new composition x * y = P(x, y)u. Then V equipped with the product * is also a Jordan algebra and is called the mutation of V with respect to u and denote by MV (u).

For u [epsilon] V, u[degrees] is an idempotent in V . It will be convenient to use the notation [M.sub.1]V (u) = V (u[degrees], 1) to indicate that the Pierce space V (u[degrees], 1) is being viewed as a subset of the Jordan algebra MV (u). Note that since V (u[degrees], 1) is itself a Jordan algebra containing u, [M.sub.1]V (u) is the mutation of V (u[degrees], 1) with respect to u and hence is a Jordan subalgebra of MV (u). Similarly [M.sub.2]V (u) will denote the orthogonal complement of V (u[degrees], 1) in V viewed as a subset of MV (u). Thus [M.sub.1]V (u) = P(u[degrees])V and [M.sub.2]V (u) = (I - P(u[degrees]))V.

Lemma 3.5.1. Let V be a Euclidean Jordan algebra of rank r with identity e and let u [epsilon] [bar.[OMEGA]](V), u [not equal to] 0, rk(u) = s. Then

(a) MV (u) = [M.sub.1]V (u)[direct sum][M.sub.2]V (u), a vector space direct sum, and for any [alpha] [epsilon] R, [M.sub.1]V (u) = P([u.sup.[alpha]])V = P([u.sup.[alpha]]) V (u[degrees], 1) and ker P([u.sup.[alpha]]) = [M.sub.2]V (u).

(b) The mapping P([u.sup.[1/2]]): MV (u) [right arrow] V is a Jordan algebra homomorphism with ker P([u.sup.[1/2]]) = [M.sub.2]V (u) and ImP([u.sup.[1/2]])=V (u[degrees], 1). Hence [M.sub.2]V (u) is an ideal in MV (u).

(c) The map P([u.sup.[1/2]])|[M.sub.1]V (u) is a Jordan algebra isomorphism from [M.sub.1]V (u) onto V (u[degrees], 1) with inverse P([u.sup.[-1/2]]).

(d) [M.sub.1]V (u) is a Euclidean Jordan algebra of rank s with identity [u.sup.+]. Further for all x [epsilon] [M.sub.1]V (u),

t[r.sub.1](x) = t[r.sub.2](P([u.sup.[1/2]])x) = t[r.sub.V](P([u.sup.[1/2]])x)

and

[det.sub.1](x) = [det.sub.2](P([u.sup.[1/2]])x) = [det.sub.V](e-P([u.sup.[1/2]])([u.sup.+]-x))

where t[r.sub.1], t[r.sub.2], [det.sub.1], [det.sub.2] are the generic traces and determinants for the Jordan algebras M[V.sub.1](u) and V (u[degrees], 1) respectively.

(e) If V is simple, so is [M.sub.1]V (u).

Remark 3.5.1. In this lemma, whenever an operator of the type P(z) (or P(x, y)) is used, it is applied with respect to the product in V, not MV (u).

For example, for x [epsilon] MV (u) we interpret P(z)x to mean 2z(zx) - [z.sup.2]x not 2z * (z * x) - (z * z) * x. Also note that by Lemma 3.3.1, P([u.sup.[alpha]]) P([u.sup.[beta]]) = P ([u.sup.[[alpha]+[beta]]]), [alpha], [beta] [member of] R.

Proof. (a) The fact that MV (u) = [M.sub.1]V (u) [direct sum] [M.sub.2]V (u) is a consequence of the Pierce decomposition of V . Also by (3.7) we have [M.sub.1]V (u) = P(u[degrees])V = P([u.sup.[alpha]]) P([u.sup.-[alpha]])V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] P([u.sup.[alpha]])V = P([u.sup.[alpha]]) P(u[degrees])V = P([u.sup.[alpha]]) V(u[degrees], 1) = P([u.sup.[alpha]]) P(u[degrees])V = P(u[degrees])P([u.sup.[alpha]])V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] P(u.sup.[degrees])V = [M.sub.1]V (u) which proves [M.sub.1]V (u) = P([u.sup.[alpha]])V = P([u.sup.[alpha]])V (u[degrees], 1). Also if x [epsilon] [M.sub.2]V (u), P([u.sup.[alpha]])x= P([u.sup.[alpha]])P(u[degrees])x = 0. Conversely, if P([u.sup.[alpha]])x = 0, then 0 = P([u.sup.-[alpha]])P([u.sup.[alpha]])x = P(u[degrees])x and therefore, x [epsilon] [M.sub.2]V(u).

(b) Let x, y [epsilon] MV (u). Then using (3.3), we have P([u.sup.[1/2]])(x*y) = P([u.sup.[1/2]])P(x, y)u = P([u.sup.[1/2]])P(x, y)P([u.sup.[1/2]])u[degrees] = P(P([u.sup.[1/2]])x, P([u.sup.[1/2]])y)u[degrees] = (P([u.sup.[1/2]])x)(P([u.sup.[1/2]])y). This last equality holds since P([u.sup.[1/2]])x and P([u.sup.[1/2]])y are in V (u[degrees], 1) and u[degrees] is the identity for V (u[degrees], 1). The rest of (b) follows from part (a).

(c) That P([u.sup.[1/2]])|[M.sub.1]V (u) is an isomorphism onto V (u[degrees], 1) follows from part

(b). Further since P([u.sup.[-1/2]]) P([u.sup.[1/2]]) = P(u[degrees]) is the identity map on [M.sub.1]V (u), P([u.sup.[-1/2]]) must be the inverse of P([u.sup.[1/2]])|[M.sub.1]V (u).

(d) Since u[degrees] is the identity for V (u[degrees], 1) and P([u.sup.[-1/2]])u[degrees] = [u.sup.+], part (c) implies [u.sup.+] is the identity for [M.sub.1]V (u). Also by Theorem 1 (vi) p. 224 of Jacobson (6), if [m.sub.x](X) is the generic minimal polynomial for x in [M.sub.1]V (u), then [m.sub.[P([u.sup.[1/2])x]]](X) is the generic minimal polynomial for P([u.sup.[1/2]])x in V (u[degrees], 1). Thus

t[r.sub.1](x) = t[r.sub.2](P([u.sup.[1/2]])x) and [det.sub.1](x) = [det.sub.2](P([u.sup.[1/2]])x).

Further since V (u[degrees], 1) is Euclidean t[r.sub.2] induces an associative inner product on V (u[degrees], 1) and hence t[r.sub.1] does the same for [M.sub.1]V (u). The rest of (d) follows from Lemma 3.3.2(a).

(e) This follows from part (c) and the fact that if V is simple so is V (u[degrees], 1).

The following example will provide a more concrete understanding of Lemma 3.5.1.

Example 3.5.1. Let V = [H.sub.r.sup.1] be the Jordan algebra of r x r Hermitian matrices over R with composition A [omicron] B = 1/2(AB + BA) and let [SIGMA] [epsilon] [H.sub.r.sup.1] be nonnegative definite. Then:

[[SIGMA].sup.+] is the Moore-Penrose inverse of [SIGMA].

[SIGMA][degrees] = [SIGMA][[SIGMA].sup.+], the orthogonal projection of [R.sup.r] onto Im[SIGMA].

[M.sub.1]V ([SIGMA]) = {A [epsilon] V: [SIGMA][degrees]A[SIGMA][degrees] = A}.

[M.sub.2]V ([SIGMA]) = {A [epsilon] V: [SIGMA][degrees]A[SIGMA][degrees] = 0}.

The product in MV ([SIGMA]) is A * B = 1/2 [A[SIGMA]B + B[SIGMA]A] and in [M.sub.1]V ([SIGMA]),

t[r.sub.1](A) = Tr([[SIGMA].sup.[1/2]]A[[SIGMA].sup.[1/2]])

and

[det.sub.1](A) = Det(I-[SIGMA][degrees] + [[SIGMA].sup.[1/2]]A[[SIGMA].sup.[1/2]]).

We are now ready to prove the key result of this section.

Theorem 3.5.2. Suppose that (1) - (5) hold:

(1) J is a Jordan algebra.

(2) J = L [direct sum] K, a vector space direct sum, where

i) L is a Jordan subalgebra of J of rank r with identity [e.sub.L] and L is simple and Euclidean;

ii) K is an ideal in J.

(3) [P.sub.L] and [P.sub.K] are the projections of J onto L and K respectively.

(4) W is a Euclidean Jordan algebra of rank s with identity [e.sub.W].

(5) [rho]: J [right arrow] W is a linear map.

Then (a) - (c) below are equivalent:

(a) [rho] is a Jordan algebra homomorphism with ker[rho] = K.

(b) There exists an integer s > 0 such that for all x [epsilon] J,

det([e.sub.w]-[rho](x)) = [det.sub.L]([e.sub.L]-[P.sub.L]x).sup.s]. (3.10)

(c) There exists an integer s > 0 such that for all x [epsilon] J and k = 1, 2, ...

tr[rho][(x).sup.k] = s t[r.sub.L][([P.sub.L]x).sup.k]. (3.11)

In the case one of (a) - (c) holds s = tr([rho]([e.sub.L]))/r . Also s = tr([rho](c)), where c is any primitive idempotent in the Jordan algebra L.

Proof. (a) [right arrow] (b): Since, by (a), [rho](x) = [rho]([P.sub.L]x) and [P.sub.L]([P.sub.L]x) = [P.sub.L]x, it suffices to show that (3.10) holds for x [epsilon] L.

Let a and b be primitive idempotents in L. By Corollary IV.2.4 of Faraut and Koranyi (4), there exists an element z [epsilon] L such that [z.sup.2] = [e.sub.L] and P(z)a = b. Now by (a), [rho](a) and [rho](b) are non-zero orthogonal idempotents in W. Also, using (3.2), we have rk([rho](b)) = tr([rho](b)) = tr([rho](P(z)a)) = tr(P([rho](z))[rho](a)) = tr([rho][(z).sup.2][rho](a)) = tr([rho]([z.sup.2])[rho](a)) = tr([rho]([e.sub.L])[rho](a)) = tr([rho]([e.sub.L]a)) = tr[rho](a).

Thus for any Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} of L, the set {[rho]([c.sub.i]): i = 1, 2, ..., r} is a family of non-zero orthogonal idempotents in W, all of the same rank, say s > 0.

Now let x = [[SIGMA].sup.r.sub.1] [[lambda].sub.j][c.sub.j] be a spectral decomposition of x in L. Then [rho](x) = [[SIGMA].sup.r.sub.1] [[lambda].sub.j][rho]([c.sub.j]) and, by Lemma 3.3.2 (b),

det([e.sub.W] - [rho](x)) = [[product].sub.1.sup.r][(1 - [[lambda].sub.j]).sup.tr([rho]([c.sub.j]))] = [[product].sub.1.sup.r][(1 - [[lambda].sub.j]).sup.s][det.sub.L][([e.sub.L] - x).sup.s].

(b) [right arrow] (c): Fix x [epsilon] J and let [rho](x) = [[SIGMA].sup.r.sub.[j=1]] [[mu].sub.j][d.sub.j] and [P.sub.L]x = [[SIGMA].sup.r.sub.[j=1]] [[lambda].sub.j]([c.sub.j]) be spectral decompositions of [rho](x) in W and [P.sub.L]x in L. Then by (3.10), we have for all [lambda] [not equal to] 0 in R,

det([e.sub.W] - [rho](1/[lambda]x)) = [det.sub.L][([e.sub.L] - [P.sub.L](1/[lambda]x)).sup.s],

or

[[product].sub.1.sup.n]([lambda] - [[mu].sub.j]) = [[lambda].sup.[n-sr]][[product].sub.1.sup.r][([lambda] - [[lambda].sub.j]).sup.s].

Thus tr[rho][(x).sup.k] = [[SIGMA].sup.n.sub.1][[mu].sub.j.sup.k] = s [[SIGMA].sup.r.sub.1][[lambda].sub.j.sup.k] = s [tr.sub.L]([P.sub.L]x).

(c) [right arrow] (a): Let x [epsilon] J. Using the spectral decompositions of [rho](x) and [P.sub.L](x) and using (3.11), it is clear that [rho](x) = 0 iff tr[rho][(x).sup.2] = 0 iff [tr.sub.L][([P.sub.L]x).sup.2] = 0 iff [P.sub.L]x = 0 iff x [epsilon] K. Thus ker [rho] = K.

Further, since K is an ideal, [x.sup.2] = [([P.sub.L]x+[P.sub.K]x).sup.2] = [([P.sub.L]x).sup.2]+z, where z [epsilon] K. Therefore to complete the proof of (a), it suffices to show that [rho]([y.sup.2]) = [rho][(y).sup.2] for y [epsilon] L.

Let y [epsilon] L and let y = [[SIGMA].sup.r.sub.[j=1]] [[lambda].sub.j][c.sub.j] be a spectral decomposition for y in L. By (3.11), tr[rho][([c.sub.j]).sup.k] = s [tr.sub.L][C.sub.j.sup.k] = s, k = 1, 2, ... and so [rho]([c.sub.j]) must be idempotent of rank s in W. Similarly, for i [not equal to] j, [rho]([c.sub.i] + [c.sub.j]) is an idempotent of rank 2s. Then, 2s = tr[rho]([c.sub.i] + [c.sub.j]) = tr([rho][([c.sub.i] + [c.sub.j]).sup.2]) = tr[([rho]([c.sub.i]) + [rho] ([c.sub.j])).sup.2] = tr([rho]([c.sub.i]) + [rho]([c.sub.j]) + 2[rho]([c.sub.i])[rho]([c.sub.j])) = 2s + 2tr([rho]([c.sub.i])[rho]([c.sub.j])). Thus tr([rho]([c.sub.i])[rho]([c.sub.j])) = 0. Hence by problem 7(c) on p. 79 of Faraut and Koranyi (4), [rho]([c.sub.i])[rho]([c.sub.j]) = 0. Therefore

[rho]([y.sup.2]) = [rho]([([r.summation over (j = 1)][[lambda].sub.j][c.sub.j]).sup.2]) = [rho]([r.summation over (j = 1)][[lambda].sub.j.sup.2][c.sub.j]) = [r.summation over (j = 1)][[lambda].sub.j.sup.2][rho]([c.sub.j]) = [([r.summation over (j = 1)][[lambda].sub.j][rho]([c.sub.j])).sup.2] = [rho][(y).sup.2],

proving (a).

Finally, in the case one of (a) - (c) holds, we may take k = 1 and x = [e.sub.L] in (3.11) to obtain s = [tr.sub.[rho]]([e.sub.L])/[tr.sub.L]([e.sub.L]). Also the proof of (a) [right arrow] (b) shows that s = tr([rho](c)) where c is any primitive idempotent in the Jordan algebra L.

Remark 3.5.2. To obtain an example of a Jordan algebra satisfying (1)-(3) of Theorem 3.5.2, consider the Jordan algebra V = [H.sub.r.sup.1] in Example 3.5.1. Then in that example take J = MV([SIGMA]), L = [M.sub.1]V([SIGMA]) and K = [M.sub.2]V([SIGMA]).

Remark 3.5.3. One might wish to investigate, in the context of ideal theory in Jordan algebra, the ideals arisen in Theorem 3.5.2..

4. Main Results. Characterization of the Wishart Distribution

We now characterize the Wishart distribution in terms of Jordan algebra representations (Theorem 4.4). This is accomplished by linking the moment generating function of the Wishart distribution with these homomorphisms via Theorem 3.5.2.

In this section [H.sub.p.sup.d], d = 1, 2, 4, will denote the simple Euclidean Jordan algebras as described in subsection 3.4, M[H.sub.p.sup.d](A) its mutation with respect to an element A [epsilon] [H.sub.p.sup.d] and [H.sub.p.sup.d](c, i), i = 0, 1, 1/2 the Pierce spaces associated with an idempotent c [epsilon] [H.sub.p.sup.d]. Lower case notation 'tr', 'det' refers to the generic trace and determinant and upper case notation 'Tr' 'Det' is the usual trace and determinant for matrices, in this case, for endomorphisms in End ([M.sub.[nxp].sup.d]) or End([R.sup.npd]). We will also make use of the functions [delta] and [phi] as described in Section 2. Also note that End([M.sub.[nxp].sup.d]) is a Euclidean Jordan algebra with identity [I.sub.n] [cross product] [I.sub.p]. Thus for T [epsilon] End([M.sub.[nxp].sup.d]), P(T) is the linear operator given by P(T)S = TST, S [epsilon] End([M.sub.[nxp].sup.d]).

We begin with some results on moment generating functions of quadratic forms.

Lemma 4.1. Let Y ~ [N.sub.k](0, [SIGMA]). Then the moment generating function of YY' is

M(t) = Det[[[I.sub.k] - 2[[SIGMA].sup.[1/2]]t[[SIGMA].sup.[1/2]]].sup.[-1/2]]

for all t [epsilon] [H.sub.k.sup.1] such that [I.sub.k] - 2[[SIGMA].sup.1/2]t[[SIGMA].sup.1/2] is positive definite.

Proof. Let Z ~ [N.sub.k](0, [I.sub.k]). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For t such that [I.sub.k] - 2[[SIGMA].sup.1/2]t[[SIGMA].sup.1/2] is positive definite, a comparison of the integrand with the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] density completes the proof.

Theorem 4.2. (a) Let Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]), [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) be a linear map and [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] the associated quadratic form. Then the moment generating function of [Q.sub.[psi]](Y) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.n] [cross product] [I.sub.P] - 2/d P([[SIGMA].sub.Y.sup.[1/2]])[psi](t) is positive definite.

(b) Let U ~ [W.sub.p.sup.d](m, [SIGMA]), [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then the moment generating function of U is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.p] - 2/d P([[SIGMA].sup.1/2])t [epsilon] [OMEGA]([H.sub.p.sup.d]). Here 'det' is the determinant in the Jordan algebra [H.sub.p.sup.d]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [delta](Y) ~ [N.sub.npd](0, 1/d[phi]([SIGMA]Y)). Lemma 4.1 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.npd] - 2/d[phi][([[SIGMA].sub.Y]).sup.1/2][phi]([psi](t))[phi][([[SIGMA].sub.Y]).sup.1/2] is positive definite. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.n] [cross product] [I.sub.p] - 2/d P([[SIGMA].sub.Y.sup.1/2])[psi](t) is positive definite.

(b) Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where Z ~ [N.sub.[mxp].sup.d] (0, [I.sub.m] [cross product] [SIGMA]) we may apply part (a) with n = m, [[SIGMA].sub.Y] = [I.sub.m] [cross product] [SIGMA] and [psi](t) = [I.sub.m] [cross product] t to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the map x [right arrow] [I.sub.m] [cross product] x is a self-adjoint representation of [H.sub.p.sup.d] on [M.sub.[mxp].sup.d] such that [I.sub.p] [right arrow] [I.sub.m] [cross product] [I.sub.p], we can apply Proposition IV.4.2 of Faraut and Koranyi (4) (with N = mpd and r = p) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.p] - 2/d P([[SIGMA].sup.1/2])t [epsilon] [OMEGA]([H.sub.p.sup.d]).

Corollary 4.3. Let Y, [psi], [Q.sub.[psi]] be as in Theorem 4.1, m [epsilon] {1, 2, 3, ...} and [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then (a) - (c) below are equivalent:

(a) [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]).

(b) For all t [epsilon] [H.sub.p.sup.d],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

(c) For all t [epsilon] [H.sub.p.sup.d] and k = 1, 2, ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

Proof. First assume [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]). Then by Theorem 4.2 (a) and (b),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

for all 2/d t [epsilon] [N.sub.0], where [N.sub.0] is a neighbourhood of 0 in [H.sub.p.sup.d]. Now (4.3) amounts to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

for all t [epsilon] [N.sub.0]. Then by analytic continuation, (4.4) holds for all t [epsilon] [H.sub.p.sup.d], proving (b).

Conversely, it is clear that (4.1) implies (4.3), which in turn (by Theorem 4.2) implies [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]).

The equivalence of parts (b) and (c) can be proved by obtaining spectral decompositions of P([[SIGMA].sub.Y.sup.1/2])[psi](t) and P([[SIGMA].sup.1/2])t and using an argument similar to that employed in the proof of (b) [right arrow] (c) in Theorem 3.5.2.

We now prove our main result.

Theorem 4.4. Suppose that:

(1) Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]).

(2) [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) is a linear map.

(3) [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] is the quadratic form associated with the linear map [psi].

(4) [rho]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d] is the linear map defined by [rho](x) = P([[SIGMA].sub.Y.sup.1/2])[psi](x). Then [Q.sub.[psi]](Y) has a Wishart distribution if and only if:

(5) There exists an element [SIGMA] [epsilon][bar.[OMEGA]]([H.sub.p.sup.d]) such that [rho] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.[nxp].sup.d] with ker [rho] = [M.sub.2][H.sub.p.sup.d]([SIGMA]). (When rk([SIGMA]) = 1 the additional condition that Tr[rho]([[SIGMA].sup.+]) is divisible by d is required.) In the case that (5) holds, [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]), where md rk([SIGMA]) = Tr[rho]([[SIGMA].sup.+]). Also md = Tr[rho](c), where c is any primitive idempotent in [M.sub.1][H.sub.p.sup.d]([SIGMA]).

Proof. First assume that [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]), [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then for x [epsilon] M[H.sub.p.sup.d]([SIGMA]), [[SIGMA].sup.+] - P([SIGMA][degrees])[epsilon] M[H.sub.p.sup.d]([SIGMA]). So by Lemma 3.5.1. (d) and Corollary 4.3, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [det.sub.i], i = 1, 2, 3 are the determinants in [M.sub.1][H.sub.p.sup.d]([SIGMA]), [H.sub.p.sup.d]([SIGMA] [degrees], 1) and [H.sub.p.sup.d] respectively. Then by Theorem 3.5.2 (with J = M[H.sub.p.sup.d]([SIGMA]), L = [M.sub.1][H.sub.p.sup.d]([SIGMA]), K = [M.sub.2][H.sub.p.sup.d]([SIGMA]), W = [End.sub.S]([M.sub.[nxp].sup.d]) and s = md), condition (5) holds.

Conversely, assume (5) holds. Since this is just condition (a) of Theorem 3.5.2 (with J, L, K and W as indicated above), we may apply the equivalent condition (b) of that Theorem together with Lemma 3.5.1 (d) to conclude that there exists an integer s = Tr[rho]([[SIGMA].sup.+])/rk([SIGMA]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [epsilon] M[H.sub.p.sup.d]([SIGMA]). Now by Lemma 3.5.1 (c), [M.sub.1][H.sub.p.sup.d]([SIGMA]) [equivalent] [H.sub.p.sup.d]([SIGMA][degrees] [equivalent] [H.sub.k.sup.d], 1), k = rk([SIGMA]). Thus if rk([SIGMA]) [greater than or equal to] 2, then there exits (by Lemmma 3.4.1) an integer m > 0 such that Tr[rho]([[SIGMA].sup.+]) = md rk([SIGMA]). Hence s = md. So by Corollary 4.3, [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d] (m, [SIGMA]). Also by Theorem 3.5.2, s = md = Tr[rho](c), where c is any primitive idempotent in [M.sub.1][H.sub.p.sup.d]([SIGMA]).

Remarks.

(a) When rk([SIGMA]) = 1, [M.sub.1][H.sub.p.sup.d]([SIGMA]) [equivalent] R for any value of d. This fact is responsible for the requirement that Tr[rho]([[SIGMA].sup.+]) be divisible by d in condition (5).

(b) Equation (2.3) provides a method for selecting the proper [SIGMA] in condition (5).

Corollary 4.5. Let (1) - (4) be as in Theorem 4.3 with [psi](x) = W [cross product] x, W [epsilon] [H.sub.n.sup.d]. Then (a) - (c) below hold:

(a) [Q.sub.[psi]](Y) = Y*WY.

(b) [rho](x) = [[SIGMA].sub.Y.sup.[1/2]] (W [cross product] x)[[SIGMA].sub.Y.sup.[1/2]].

(c) [Q.sub.[psi]](Y) follows a Wishart distribution if and only if there exists an element [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]) such that

(i) ker [rho] = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} and

(ii) for all x [epsilon] [H.sub.p.sup.d],

[[SIGMA].sub.Y.sup.[1/2]](W [cross product] x[SIGMA]x)[[SIGMA].sub.Y.sup.[1/2]] = [[SIGMA].sub.Y.sup.[1/2]](W [cross product] x)[[SIGMA].sub.Y](W[cross product]x)[[SIGMA].sub.Y.sup.[1/2]]. (4.5)

(When rk([SIGMA]) = 1, the additional condition that Tr [rho]([[SIGMA].sup.+]) is divisible by d is required).

In the case (i) and (ii) hold, md rk([SIGMA]) = Tr[rho]([[SIGMA].sup.+]).

Proof. Part (a) was shown in Example 2.1 and part (b) follows from the definition of the operator P([[SIGMA].sub.Y.sup.1/2]). Further, part (c) follows from Theorem 4.4 on noting that by Lemma 3.5.1 (a), [M.sub.2][H.sub.p.sup.d]([SIGMA]) = kerP([SIGMA]) = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} and that equation (4.5) is simply the equation [rho](x * x) = [rho](x[SIGMA]x) = [rho][(x).sup.2].

To illustrate the ideas in Theorem 4.4 and Corollary 4.5, we provide some simple examples.

Example 4.1. In the illustrations below, Y = U + iV ~ [N.sub.2x2.sup.2](0, [[SIGMA].sub.Y]) and [psi]: [H.sub.2.sup.2] [right arrow] [End.sub.S]([M.sub.2x2.sup.2]), [psi](t) = [I.sub.2] [cross product] t. Also, [E.sub.ij] represents the 2 x 2 matrix whose (i, j)th entry is 1 and all other entries 0.

(a) [[SIGMA].sub.Y] = [E.sub.11] [cross product] [E.sub.11].

In this case [Q.sub.[psi]](Y) = Y*Y = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking [rho] as in Corollary 4.5, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Choose [SIGMA] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [epsilon] [H.sub.2.sup.2]. Then [rho]: M[H.sub.2.sup.2]([SIGMA]) [right arrow] End([M.sub.2x2.sup.2) is a self-adjoint Jordan algebra homomorphism with ker [rho] = [M.sub.2][H.sub.2.sup.2]([SIGMA]). Here r = rk([SIGMA]) = 1, d = 2 and Tr([rho]([[SIGMA].sup.+])) = 2. Thus by Corollary 4.5, Y*Y ~ [W.sub.[2x2].sup.2](1, [SIGMA]).

(b) [[SIGMA].sub.Y] is such that for X = A + iB [epsilon] [M.sub.[2x2].sup.2], [[SIGMA].sub.Y] (X) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case, [Q.sub.[psi]](Y) = Y*Y = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for t [epsilon] [H.sub.2.sup.2] the map [rho](t) is given by [rho](t)(A + iB) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking [SIGMA] [epsilon] [H.sub.2.sup.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we see that [rho] is a self-adjoint representation of M[H.sub.2.sup.2]([SIGMA]) on [M.sub.[2x2].sup.2]. (Note that for t [epsilon] [H.sub.2.sup.2], [t.sub.11] [epsilon] R). Here r = rk([SIGMA]) = 1, d = 2 and Tr([rho]([[SIGMA].sup.+])) = 1. Since d does not divide Tr([rho]([[SIGMA].sup.+])), Y*Y is not [W.sub.[2x2].sup.2](m, [SIGMA]). Note however that if we view Y as taking values in [M.sub.2x2.sup.1] and set Z = [square root of (dY)] = [square root of (2Y)], we have Z ~ [N.sub.2x2.sup.1](0, [[SIGMA].sub.Y]) and Z*Z ~ [W.sub.2x2.sup.1](m, [SIGMA]).

Corollary 4.6. Let Y ~ [N.sub.nxp.sup.d](0, A [cross product] [SIGMA]), A [cross product] [SIGMA] [not equal to] 0, W [epsilon] [H.sub.n.sup.d] with AW A [not equal to] 0 and Q(Y) = Y*WY. Then Q(Y) follows a Wishart distribution if and only if

AW AW A = AW A. (4.6)

(When rk([SIGMA]) = 1, the additional condition that Re Tr(AW) be divisible by d is required). In the case (4.6) holds, Q(Y) ~ [W.sub.p.sup.d](m, [SIGMA]), where m = Re Tr(AW).

Proof. First suppose rk([SIGMA]) [greater than or equal to] 2. Let [rho] be as in Corollary 4.5. Then

[rho](x) = [A.sup.1/2]W[A.sup.1/2][cross product][[SIGMA].sup.1/2]x[[SIGMA].sup.1/2].

Since AW A [not equal to] 0, ker [rho] = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0}. Further, for this particular [rho], (4.6) is equivalent to (4.5). Thus the first part of this result follows from Corollary 4.5. Also if (4.6) holds, Corollary 4.5 and Lemma 2.1 yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus m = Re Tr(AW).

Remark 4.2. Let g be a Hilbert space isomorphism from [M.sub.[nxp].sup.d] onto a Hilbert space F. Then the map

T [right arrow] [T.sub.g] = gT[g.sup.-1]

is an algebra isomorphism from End([M.sub.nxp.sup.d]) onto End(F). Let [psi] and [rho] be as in Theorem 4.4 and define [[psi].sub.g] and [[rho].sub.g] by

[[psi].sub.g](x) = [([psi](x)).sub.g] and [[rho].sub.g](x) = [([rho](x)).sub.g].

Then [[psi].sub.g], [[rho].sub.g]: [H.sub.p.sup.d] [right arrow] [End.sub.S](F) and

[[rho].sub.g](x) = P([([[SIGMA].sub.Y.sup.1/2]).sub.g])[[psi].sub.g](x).

Further, [rho] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.nxp.sup.d] if and only if [[rho].sub.g] is a self-adjoint representation of M[H.sub.p.sup.d]([SIGMA]) on F. Also ker [rho] = ker [[rho].sub.g]. Thus in Theorem 4.4 (Corollary 4.5), it may be easier verify condition (5) (conditions (i) and (ii)) by making a judicious choice for g and using [[psi].sub.g] and [[rho].sub.g] in place of [psi] and [rho].

In particular, if one takes F = [M.sub.pxn.sup.d] and g(X) = X*, X [epsilon] [M.sub.nxp.sup.d], then for all A [epsilon] [M.sub.nxn.sup.d] and B [epsilon] [M.sub.pxp.sup.d],

[(A [cross product] B).sub.g] = B [cross product] A.

Also,

[([[SIGMA].sub.Y]).sub.g] = [[SIGMA].sub.Y*].

Thus in Corollary 4.5, one may replace [rho](x) and [psi](x) by

[[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2] (x [cross product] W)[[SIGMA].sub.Y*.sup.1/2] and [[psi].sub.g](x) = x [cross product] W.

The corollary below is an application of the above principle.

Corollary 4.7. Let Y ~ [N.sub.nxp.sup.d](0, [[SIGMA].sub.Y]). Suppose that [[SIGMA].sub.Y*] = [[SIGMA].sup.r.sub.i=1] [E.sub.ii] [cross product] [A.sub.i], r [less than or equal to] p, where [E.sub.ii] is the p x p matrix whose (i, i)th entry is 1 and all other entries 0, [A.sub.i] [epsilon] [OMEGA]([H.sub.n.sup.d]) and W [epsilon] [H.sub.n.sup.d] with Re Tr([A.sub.i]W) > 0. Then Y* WY follows a Wishart distribution if and only if (1): (1) There exist real numbers [[sigma].sub.k] > 0, k = 1, 2, ..., r such that for all i, j, k,[A.sub.i]W [A.sub.k]W [A.sub.j] = [[sigma].sub.k][A.sub.i]W [A.sub.j]. (When r = 1, the additional condition that 1/[[sigma].sub.1] ReTr([A.sub.i]W) be divisible by d is required.)

In case that (1) holds, Y*WY ~ [W.sub.p.sup.d](m, [SIGMA]), where

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii] and mr = [r.summation over (i = 1)][1/[[sigma].sub.i]]Tr([A.sub.i]W).

Proof. If r = 1, [[SIGMA].sub.Y*] = [E.sub.11] [cross product] [A.sub.1]. So [[SIGMA].sub.Y] = [A.sub.1] [cross product] [E.sub.11]. Letting Z = [square root of ([[sigma].sub.1]Y)] and noting that [[SIGMA].sub.Z] = [A.sub.1] [cross product] [[sigma].sub.1][E.sub.11] and that Y*WY is Wishart if and only if Z*WZ is, the result follows from Corollary 4.6.

Now let r > 1. First assume Y*WY ~ [W.sub.p.sup.d] (m, [SIGMA]). By Corollary 4.5 and Remark 4.2, the map

[[rho].sub.g]: M[H.sub.p.sup.d]([SIGMA])[right arrow]Ends([M.sub.pxn.sup.d]), [[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2](x[cross product]W)[[SIGMA].sub.Y*.sup.1/2]

is a Jordan algebra homomorphism and md rk([[SIGMA].sup.+]) = Tr[[rho].sub.g]([[SIGMA].sup.+]). In this case,

[[rho].sub.g](x) = [[SIGMA].sub.i,j = 1.sup.r] [x.sub.ij][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.j.sup.1/2]. (4.7)

We first determine the nature of [SIGMA]. By Lemma 2.1 with (2.4) and (2.3), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.i] = ([A.sub.i], W) = Re Tr([A.sub.i]W) > 0. Thus

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii], [[sigma].sub.i] = [[a.sub.i]/m]>0.

Now in the Jordan algebra M[H.sub.p.sup.d]([SIGMA]),

x * x = x[SIGMA]x = [summation over (k = 1)][[sigma].sub.k][x.sub.ik][x.sub.kj][E.sub.ij]. (4.8)

Hence by (4.7),

[[rho].sub.g](x * x) = [r.summation over (i,j = 1)][r.summation over (k = 1)][[sigma].sub.k][x.sub.ik][x.sub.kj][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.j.sup.1/2] (4.9)

and

[[rho].sub.g][(x).sup.2] = [r.summation over (i,j = 1)][r.summation over (k = 1)][x.sub.ik][x.sub.kj][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.k]W[A.sub.j.sup.1/2]. (4.10)

Since [[rho].sub.g](x*x) = [[rho].sub.g][(x).sup.2], a comparison of (4.9) and (4.10) yields [A.sub.i.sup.1/2] W [A.sub.k]W [A.sub.j.sup.1/2] = [[sigma].sub.k][A.sub.i.sup.1/2] W [A.sub.j.sup.1/2] which proves (1).

Conversely, suppose that (1) holds. Define [SIGMA] by

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii].

Then (4.7) - (4.10) hold.

Condition (1) together with (4.9) and (4.10) yields [[rho].sub.g](x * x) = [[rho].sub.g][(x).sup.2].

Also, as Re Tr([A.sub.i]W) > 0, [A.sub.i.sup.1/2]W [A.sub.i.sup.1/2] [not equal to] = 0. Thus by (1), 0 [not equal to] [A.sub.i]W [A.sub.i] = [A.sub.i]W [A.sub.j]W [A.sub.i] and so [A.sub.i]W [A.sub.j] [not equal to] 0. Therefore by (4.7),

ker [[rho].sub.g] = {x[epsilon][H.sub.p.sup.d]: [x.sub.ij] = 0,i,j = 1,2, ... r} = {x[epsilon][H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} = [M.sub.2][H.sub.p.sup.d]([SIGMA]).

Thus by Corollary 4.5, Y*WY ~ [W.sub.p.sup.d] (m, [SIGMA]), where md rk([SIGMA]) = Tr[[rho].sub.g]([[SIGMA].sup.+]). (Here rk([SIGMA]) = r > 1). Since [[SIGMA].sup.+] = [[SIGMA].sup.r.sub.i=1] 1/[[sigma].sub.i] [E.sub.ii], [[rho].sub.g]([[SIGMA].sup.+]) = [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] [E.sub.ii] [cross product] [A.sub.i.sup.[1=2]]W [A.sub.i.sup.[1=2]]. Then by Lemma 2.1, Tr[[rho].sub.g]([[SIGMA].sup.+]) = d [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] Re Tr([A.sub.i]W). Hence we have mr = [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] Re Tr([A.sub.i]W).

The following example gives an application of Corollary 4.7 and also provides an illustration of a quadratic form Y*WY that is Wishart but W is not nonnegative definite.

Example 4.2. Let W [epsilon] [H.sub.3.sup.1] and Y ~ [N.sub.[3x2].sup.1] (0, [[SIGMA].sub.Y]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [[SIGMA].sub.Y*] = [[SIGMA].sup.2.sub.i=1] [E.sub.ii] [cross product] [A.sub.i]. So Corollary 4.7 applies. It is easily verified that Tr[A.sub.i]W > 0, i = 1, 2 and that [A.sub.i]W [A.sub.k]W [A.sub.j] = [[sigma].sub.k][A.sub.i]W [A.sub.j], where [[sigma].sub.1] = 4 and [[sigma].sub.2] = 1. Thus Y*WY ~ [W.sub.2.sup.1] (m, [SIGMA]), where m = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Further, letting x = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the Jordan algebra homomorphism [[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2](x [cross product] W) [[SIGMA].sub.Y*.sup.1/2] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now prove a general version of Cochran Theorem (Cochran (2)).

Lemma 4.8. Suppose that:

(1) Y ~ [N.sub.nxp.sup.d] (0, [[SIGMA].sub.Y]);

(2) [[psi].sub.i]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.nxp.sup.d]), i [epsilon] I are linear maps;

(3) [Q.sub.i]:[M.sub.nxp.sup.d] [right arrow] [H.sub.p.sup.d], i [epsilon] I are the quadratic forms associated with the linear maps [[psi].sub.i];

and

(4) [[rho].sub.i]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.nxp.sup.d]); i [epsilon] I are the linear maps defined by [[rho].sub.i](x) = P([[SIGMA].sub.Y.sup.[1=2]]) [[psi].sub.i](x).

Then {[Q.sub.i](Y)} is an independent family if and only if

(5) for any distinct i, j [epsilon] I and any u, v [epsilon] [H.sub.p.sup.d], [[rho].sub.i](u)[[rho].sub.j](v) = 0.

Proof. Suppose that {[Q.sub.i](Y)} is an independent family. Let i [not equal to] j and u, v [epsilon] [H.sub.p.sup.d]. Then (u, [Q.sub.i](Y)) = (Y, [[psi].sub.i](u)Y) and (v, [Q.sub.j](Y)) = (Y, [[psi].sub.j](v)Y) are independent. Since

<Y,[[psi].sub.i](u)Y> = <[delta](Y),[phi]([[psi].sub.i](u))[delta](Y)> = [delta](Y)'[phi]([[psi].sub.i](u))[delta](Y)

and [delta](Y) ~ [N.sub.npd](0, 1/d[phi]([[SIGMA].sub.Y])), Theorem 4s p.71 of Searle (18) gives

[1/[d.sup.3]][phi]([[SIGMA].sub.Y])[phi]([[psi].sub.i](u))[phi]([[SIGMA].sub.Y])[phi]([[psi].sub.j](v))[phi]([[SIGMA].sub.Y]) = 0

which is equivalent to condition (5).

Conversely, suppose that (5) holds. We may assume that I = {1, 2 ..., l}. Let Q(Y) = ([Q.sub.i](Y)). To show that {[Q.sub.i](Y)} is an independent family, it suffices to show that [M.sub.Q(Y)](t) = [[product].sup.l.sub.[i=1]] [M.sub.[Q.sub.i](Y)]([t.sub.i]) for t = ([t.sub.i]) in [N.sub.0], where [N.sub.0] is a neighbourhood of 0 in H = [H.sub.p.sup.d] x [H.sub.p.sup.d] x ... x [H.sub.p.sup.d] (l times). Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus by Lemma 4.1,

[[M.sub.Q(Y)](t) = Det [[I.sub.npd] - 2/d [phi]([[[sigma].sub.Y].sup.1/2] ([l.summation over (I=1)] [phi] ([[psi].sub.i]([t.sub.i]))) [phi][([[sigma].sub.Y]).sup.1/2]].sup.1/2] = Det [[[I.sub.n] [cross product] [I.sub.p] - 2/d P [([[SIGMA].sub.Y]).sup.1/2] [l.summation over (I=1)] [[psi].sub.i] [t.sub.i]].sup.-1/2].

Then by condition (5),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 4.9. Let E be a real vector space. Suppose that S, T [epsilon] End(E) are such that [S.sup.2] = S and 1/2 (ST + TS) = T. Then ST = TS.

Proof. Multiplying both sides of 2T = ST + TS on the left by S and then on the right by S yields 2ST = [S.sup.2]T + STS and 2TS = STS + T[S.sup.2]. Since [S.sup.2] = S, ST = STS and TS = STS whence ST = TS.

Lemma 4.10. (a version of Cochran Theorem). Let (1) - (4) be as in Lemma 4.8 and let [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then [{[Q.sub.i](Y)}.sub.i[epsilon]I] is an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables for some [m.sub.i] [epsilon] {1, 2, ...} if and only if (a) and (b) below hold. (a) For all i [epsilon] I, [[rho].sub.i] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.nxp.sup.d] with ker [[rho].sub.i] = [M.sub.2][H.sub.p.sup.d]([SIGMA]) (If rk([[SIGMA].sup.+]) = 1, we also require that Tr[[rho].sub.i]([[SIGMA].sup.+]) is divisible by d.)

(b) For all i [not equal to] j in I, [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.j]([[SIGMA].sup.+]) = 0. In case that (a) and (b) hold, [m.sub.i]d rk([[SIGMA].sup.+]) = Tr [[rho].sub.i]([[SIGMA].sup.+]).

Proof. Let {[Q.sub.i](Y)} be an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables. Then (a) and (b) hold by Theorem 4.4 and Lemma 4.8.

Conversely, suppose that (a) and (b) hold. By Theorem 4.4, [Q.sub.i](Y) ~ [W.sub.p.sup.d] ([m.sub.i], [SIGMA]). Now let u, v [epsilon] [H.sub.p.sup.d]. Then u = [u.sub.1]+[u.sub.2] and v = [v.sub.1]+[v.sub.2], [u.sub.1], [v.sub.1] [epsilon] [M.sub.1][H.sub.p.sup.d]([SIGMA]), [u.sub.2], [v.sub.2] [epsilon] [M.sub.2][H.sub.p.sup.d]([SIGMA]) and for i [not equal to] j in I,

[[rho].sub.i](u)[[rho].sub.j](v) = [[rho].sub.i]([u.sub.1])[[rho].sub.j]([v.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+] * [u.sub.1]) [[rho].sub.j]([[SIGMA].sup.+] * [v.sub.1]). (4.11)

Since [[rho].sub.i]([[SIGMA].sup.+]) = [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.i]([[SIGMA].sup.+]) and [[rho].sub.i]([u.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+] * [u.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+]) [omicron] [[rho].sub.i]([u.sub.1]) = 1/2 [[[rho].sub.i]([[SIGMA].sup.+]) [[rho].sub.i]([u.sub.1])+[[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+])], Lemma 4.9 gives [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.i]([u.sub.1]) = [[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+]). Similarly [[rho].sub.j]([[SIGMA].sup.+]) [[rho].sub.j]([v.sub.1]) = [[rho].sub.j]([v.sub.1])[[rho].sub.j]([[SIGMA].sup.+]). Thus [[rho].sub.i]([[SIGMA].sup.+]*[u.sub.1])[[rho].sub.j]([[SIGMA].sup.+]*[v.sub.1]) = [[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+]) [[rho].sub.j]([[SIGMA].sup.+])[[rho].sub.j]([v.sub.1]) = 0. Hence by (4.11), [[rho].sub.i](u)[[rho].sub.j](v) = 0 and hence by Lemma 4.8, [[Q.sub.i](Y)} is an independent family.

Corollary 4.11. Let Y ~ [N.sub.nxp.sup.d] (0, A [cross product] [SIGMA]), A [cross product] [SIGMA] [not equal to] 0, [W.sub.i] [epsilon] [H.sub.n.sup.d] with A[W.sub.i]A [not equal to] 0, i [epsilon] I and [Q.sub.i](Y) = Y*[W.sub.i]Y. Then {[Q.sub.i](Y)} is an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables if and only if (a) and (b) below hold.

(a) For all i [epsilon] I, A[W.sub.i]A[W.sub.i]A = A[W.sub.i]A. (In the case rk([SIGMA]) = 1, we also require that Re Tr([A.sub.i]W) be divisible by d.)

(b) For all i [not equal to] j in I, A[W.sub.i]A[W.sub.j]A = 0. In case that (a) and (b) hold, [m.sub.i]d rk([SIGMA]) = Tr [[rho].sub.i]([[SIGMA].sup.+]).

Proof. The desired results follow from Theorem 4.10 and Corollary 4.6 on noting that in this case [[psi].sub.i](x) = [W.sub.i] [cross product] x and [[rho].sub.i](x) = [A.sup.1/2][W.sub.i][A.sup.1/2] [cross product] [[SIGMA].sup.1/2]x[[SIGMA].sup.1/2].

Acknowledgement

The authors would like to thank the referee for his careful reading of this paper and for his comments which led to the present improved version.

References

(1) H. Braun, and M. Koecher, Jordan Algebren, Springer Verlag, Berlin-Heidelberg, 1966.

(2) W. G. Cochran, The distribution of quadratic forms in a normal system with applications to the analysis of covariance, Proc. Cambridge Philos. Soc. 30(1934), 178-191.

(3) M. C. M. Degunst, On the distribution of general quadratic functions in normal vectors, Statist. Neerlandica 41(1987), 245-251.

(4) J. Faraut, and A. Kor_anyi, Analysis on Symmetric Cones. Oxford University Press, 1994.

(5) N. R. Goodman, Statistical analysis based on a certain multivariate complex Gaussian distribution (An introduction), Ann. Math. Statist. 34(1963), 152-176.

(6) N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Coll. Publ. XXXIX(1968), Providence, Rhode Island.

(7) S. J. Jensen, Covariance hypotheses which are linear in both the covariance and inverse covariance, Ann. Statist. 16(1988), 302-322.

(8) C. G. Khatri, Quadratic forms in normal variables, 443-469, Handbooks of Statistics, P. R. Krishnaiah, Ed., North Holland, Amsterdam, 1980.

(9) G. Letac, and H. Massam, Quadratic and inverse regressions for Wishart distributions, Ann. Statist. 26(1998), 573-595.

(10) J. D. Malley. Optimal unbiased estimation of variance components. Lecture notes in statistics, Springer-Verlag, New York, 1986.

(11) J. Masaro and C. S. Wong, Wishart distributions associated with matrix quadratic forms, J. Multivariate Anal. 85(2003), 1-9.

(12) J. Masaro and C. S. Wong, Laplace-Wishart distributions associated with matrix quadratic forms, Hawaii International Conference on Statistics, Mathematics and Related Fields, January 16-19, 2006.

(13) H. Massam, An exact decomposition theorem and a unified view of some related distributions for a class of exponential transformation models on symmetric cones, Ann. Statist. 22(1994), 369-394.

(14) H. Massam and E. Neher, On transformations and determinants of Wishart variables on symmetric cones, J. Theoret. Probab. 10(1997), 867-902.

(15) H. Massam and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras. Ann. Statist. 26(1998), 1051-1082.

(16) T. Mathew and K. Nordstrom, Wishart and chi-square distributions associated with matrix quadratic forms, J. Multivariate Anal. 61(1997), 129-143.

(17) R. J. Pavur, Distribution of multivariate quadratic forms under certain covariance structures, Canad. J. Statist. 15(1987), 169-176.

(18) S. R. Searle, Linear Models, Wiley, New York, 1971.

(19) J. Seely, Quadratic subspaces and completeness, Ann. Math. Statist. 42(1971), 710-721.

(20) C. S. Wong, J. Masaro, and T. Wang, Multivariate versions of Cochran's theorems, J. Multivariate Anal. 39(1991), 154-174.

(21) C. S. Wong and T. Wang, Multivariate versions of Cochran's Theorems II, J. Multivariate Anal. 44(1993), 146-159.

(22) C. S.Wong and J. Masaro, Multivariate Versions of Cochran Theorems via Jordan Algebra Homomorphisms, in an invited talk on Algebraic Methods in Statistics at Fields Institute, Toronto, October 25-30, 1999.

Joe Masaro [dagger]

Acadia University, Wolfville, Nova Scotia, Canada B4P 2R6

and

Chi Song Wong [double dagger]

University of Windsor, Windsor, Ontario, Canada N9B 3P4

Received June 16, 2008, Accepted Oecember 16, 2008.

* AMS 1991 Mathematics Subject Classification. Primary 62H05; secondary 62H10.

[dagger] E-mail: joe.masaro@acadiau.ca

[double dagger] E-mail: cswong@uwindsor.ca

This paper is a sequel to Masaro and Wong (12) and, like (12), is based on Wong and Masaro (22).

Consider a normally distributed random matrix Y with mean 0 and general covariance matrix [[SIGMA].sub.Y]. This paper deals with the following question.

(i) When does a quadratic form in Y, Q(Y), follow a Wishart distribution with m degrees of freedom and scale parameter [SIGMA]?

In the case where Y is a real random matrix and Q(Y) = Y'WY with W nonnegative definite question (i) has been addressed by various authors; see for example, Khatri (8), DeGunst (3), Pavur (17), Wong and Wang (21) and Mathew and Nordstrom (16). For the most general case, i.e. W is symmetric and [[SIGMA].sub.Y] and [SIGMA] are nonnegative definite, Wong, Masaro and Wang (20) obtained some results by placing restrictions on the image space of [[SIGMA].sub.Y]. These restrictions were lifted in Masaro and Wong (11) but the necessary and sufficient conditions obtained for the Wishartness of Y'WY were quite complicated. It was then discovered (Wong and Masaro (22) and Masaro and Wong (12)) that these conditions reflected the fact that the Wishartness of Y'WY depended on whether or not a certain linear transformation was a Jordan algebra homomorphism.

We shall exploit the connection between Jordan algebras and the Wishart distribution, and characterize the Wishartness of a quadratic form Q(Y) in terms of Jordan algebra homomorphisms or more precisely, Jordan algebra representations. The framework of Jordan algebras provides a unified approach to question (i) in that the cases where Y is a real, complex or quaternionic normal random matrix can be dealt with simultaneously as part of a general theory. In addition we feel that this point of view provides a deeper insight into the nature of the Wishart distribution.

The outline of this paper is as follows. In Section 2 we set up some notation and recall the definitions of the normal and Wishart distributions. Section 3 summarizes the results from the theory of Jordan algebras that we will need. The key result of this section is Theorem 3.5.2 which links the concept of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution. In Section 4 we give our main results (Theorem 4.4 and Corollary 4.5) which characterize the Wishartness of Q(Y).

We remark that the applications of Jordan algebras to statistics is not new. See Seely (19), Malley (10), Jensen (7), Massam (13), Massam and Neher (14), (15) and Letac and Massam (9).

2. Preliminaries

Let [A.sub.d] denote R, C or H according to d = 1, 2 or 4, where R denotes the field of real numbers, C the field of complex numbers and H the division ring of quaternions over R. Each x [member of] H may be represented as x = [x.sub.1]+i[x.sub.2]+j[x.sub.3]+k[x.sub.4], where [x.sub.i] [member of] R, [i.sup.2] = [j.sup.2] = [k.sup.2] = -1, ij = -ji = k, jk = -kj = i and ki = -ik = j. The conjugate of x is [bar.x] = [x.sub.1]-i[x.sub.2]-j[x.sub.3]-k[x.sub.4] and the real part of x is Re x = [x.sub.1]. Analogous definitions apply to C. Note that multiplication in H is associative but not commutative and that for x, y [Note that multiplication] H, [bar.xy] = [bar.y] [bar.x].

Let [M.sub.[nxp].sup.d] denote the family of n X p matrices over [A.sub.d]. A matrix A [member of] [M.sub.[nxp].sup.4] may be written as A = [A.sub.1] + i[A.sub.2] + j[A.sub.3] + k[A.sub.4], where [A.sub.i] [member of] [M.sub.[nxp].sup.1]. The transpose, conjugate and adjoint of A are defined by A' = [A'.sub.1] + i[A'.sub.2] + j[A'.sub.3] + k[A'.sub.4], [bar.A] = [A.sub.1]-i[A.sub.2]-j[A.sub.3]-k[A.sub.4] and A* = [bar.A'] respectively. (Here [A'.sub.i] is the usual matrix transpose for matrices over R). Similar notions apply to [M.sub.[nxp].sup.d] for d = 1, 2. Of course for A [member of] [M.sub.[nxp].sup.1], [bar.A] = A, A* = A' and Re A = A.

We remark that the familiar formulas (AB)' = B'A' and [bar.AB] = [bar.A] [bar.B] hold only in the cases d = 1 or 2 but the formulas (AB)* = B* A* and Re Tr(AB) = Re Tr(BA) hold for d = 1, 2 or 4.

A matrix A [member of] [M.sub.[nxn].sup.d] is called Hermitian if A* = A. The family of n x n Hermitian matrices over [A.sub.d] will be denoted by [H.sub.n.sup.d].

We shall view [M.sub.[nxp].sup.d] as a Euclidean vector space, i.e., a vector space over R with the inner product (A, B) = ReTr(AB*). Thus the dimension of [M.sub.[nxp].sup.d] is npd. Also we have (A, B) = ([bar.A], [bar.B]) and (A, BC) = (B* A, C) = (AC*, B) whenever the matrix multiplication is defined.

Let End([M.sub.[nxp].sup.d]) denote the (real) vector space of endomorphisms of [M.sub.[nxp].sup.d]. The adjoint of T [member of] End([M.sub.[nxp].sup.d]) is denoted by T*; so for all X, Y [member of] [M.sub.[nxp].sup.d], (T(X), Y) = (X, T*(Y)). T is called self-adjoint of T* = T. The space of self-adjoint endomorphisms will be denoted by [End.sub.S]([M.sub.[nxp].sup.d]). For A [member of] [M.sub.[nxn].sup.d] and B [member of] [M.sub.[pxp].sup.d], the Kronecker product A [cross product] B is defined as the element in End([M.sub.[nxp].sup.d]) such that

(A [cross product] B)(C) = ACB*.

For X [member of] [M.sub.[nxp].sup.d], let [delta](X) (or [X]) be the coordinate vector of X in [R.sup.npd] with respect to an orthogonal basis for [M.sub.[nxp].sup.d] and for T [member of] End([M.sub.[nxp].sup.d]), let [phi](T) (or [T]) be the matrix representation with respect to this basis. Thus (X, Y) = ([delta](X), [delta](Y)) and [delta](T(X)) = [phi](T)[delta](X). Also [phi](T*) = [phi](T)' and T is self-adjoint (nonnegative definite, positive definite) according as [phi](T) is symmetric (nonnegative definite, positive definite).

We shall view End([M.sub.[nxp].sup.d]) as a real inner product space with inner product

(S, T) = ([phi](S), [phi](T)) = Tr([phi](S)[phi](T)').

One may also write (S, T) = Tr(ST*) since by definition, Tr(ST*) = Tr [phi](ST*) = Tr([phi](S)[phi](T)').

The following lemma will be useful.

Lemma 2.1. Let A, C [member of] [M.sub.[nxn].sup.d] and B, D [member of] [M.sub.[pxp].sup.d]. Then:

(a) (A [cross product] B, C [cross product] D) = d(A, C)(B,D).

(b) Tr(A [cross product] B) = d Re Tr(A)Re Tr(B).

Proof. We shall merely prove the result for the case d = 4.

(a) Let B be the standard orthogonal basis for [M.sub.[nxp].sup.d]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Now each basis element in B is of the form F = [alpha][F.sub.rs], where [alpha] = 1, i, j or k and [F.sub.rs] is the n x p matrix with (r, s)th entry equal to 1 and all other entries 0. Thus

F* CA* F = [bar.[alpha]][F.sub.sr]CA* [alpha][F.sub.rs] = [bar.[alpha]][u.sub.rr][alpha][E.sub.ss]

where [u.sub.rr] is the (r, r)th entry of CA* and [E.sub.ss] is the p x p matrix with (s, s)th entry equal to 1 and other entries 0. Then using the fact that [[SIGMA].sub.[alpha]] [bar.[alpha]][u.sub.rr][alpha] = d Re [u.sub.rr] (d = 4 here), we have

[summation over (F [member of] B)]F*CA*F = [summation over (s)][summation over (r)][summation over ([alpha])][bar.[alpha]][u.sub.rr][alpha][E.sub.ss] = d Re Tr(CA*)[I.sub.p].

Thus by (2.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) Since Tr(A [cross product] B) = (A [cross product] B, [I.sub.n] [cross product] [I.sub.p]), part (b) follows by taking C = [I.sub.n] and D = [I.sub.p] in (a).

Let [N.sub.k]([gamma], [SIGMA]) denote the usual normal distribution over [R.sup.k] with mean [gamma] and nonnegative definite covariance matrix [SIGMA]. A random variable Y taking values in [M.sub.[nxp].sup.d], d = 1, 2 or 4, is said to have a real, complex or quaternionic normal distribution with mean [[mu].sub.Y] [member of] [M.sub.[nxp].sup.d] and covariance matrix [[SIGMA].sub.Y] [member of] [End.sub.S]([M.sub.[nxp].sup.d]) if [delta](Y) ~ [N.sub.npd]([delta]([[mu].sub.Y]), 1/d[phi]([[SIGMA].sub.Y])). In this case, we write Y ~ [N.sub.[nxp].sup.d]([mu], [[SIGMA].sub.Y]). Note that [N.sub.k]([mu], [SIGMA]) = [N.sub.[kx1].sup.1]([mu], [SIGMA] [cross product] 1).

A random variable U taking values in [H.sub.p.sup.d], d = 1, 2, or 4, is said to have a real, complex or quaternionic Wishart distribution with m degrees of freedom and scale matrix [SIGMA] [member of] [H.sub.p.sup.d] if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where Z ~ [N.sub.[mxp].sup.d](0, [I.sub.m] [cross product] [SIGMA]). In this case, we write U ~ [W.sub.p.sup.d] (m, [SIGMA]).

Remark 2.1. Usually, in the case d = 2, one defines the [W.sub.p.sup.2] (n, [SIGMA]) distribution to be the distribution of [n.summation over (i=1)] [Y.sub.i][Y.sub.i]*, where the [Y.sub.i]'s are iid [N.sub.[px1].sup.2](0, [SIGMA] [cross product] 1) (see Goodman (5)). Note that one may also write [n.summation over (i=1)] [Y.sub.i][Y.sub.i]* as Y'[bar.Y], where Y' = [[Y.sub.1], [Y.sub.2], ..., [Y.sub.n]]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then Z*Z = Y'[bar.Y] and since the [Y.sub.i]*'s are iid [N.sub.[1xp].sup.2](0, 1 [cross product] [SIGMA]), Z ~ [N.sub.[nxp].sup.2](0, [I.sub.n] [cross product] [SIGMA]). Let [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) be a linear map.

For each y [member of] [M.sub.[nxp].sup.d], the linear form on the real vector space [H.sub.p.sup.d] defined by t [right arrow] (y, [psi](t)y) is given by an inner product on [H.sub.p.sup.d]. (For simplicity, [psi](t)(y) is abbreviated as [psi](t)y.) Thus there is an element in [H.sub.p.sup.d] depending y and [psi], call it [Q.sub.[psi]](y), such that

(y, [psi](t)y) = (t, [Q.sub.[psi]](y)) (2.2)

for all t [member of] [H.sub.p.sup.d]. We call the map [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] the [H.sub.p.sup.d]-valued quadratic form associated with the linear map [psi].

If Y ~ [N.sub.[nxp].sup.d] (0, [[SIGMA].sub.Y]), then [Q.sub.[psi]](Y) is a random quadratic form taking values in [H.sub.p.sup.d]. The mean of [Q.sub.[psi]](Y) can be obtained as follows:

(t, E([Q.sub.[psi]](Y))) = [1/d]([[SIGMA].sub.Y],[psi](t)). (2.3)

Indeed, (t, E([Q.sub.[psi]](Y))) = E(t,[Q.sub.[psi]](Y)) = E(Y, [psi](t)(Y))=E([delta](Y), [phi]([psi](t))[delta](Y)) = E([delta](Y)[delta](Y)', [phi]([psi](t))) = (1/d[phi]([[SIGMA].sub.Y]), [phi]([psi](t))) = 1/d ([[SIGMA].sub.Y]), [psi](t)).

The main purpose of this paper is to determine necessary and sufficient conditions for [Q.sub.[psi]](Y) to have a Wishart distribution. These conditions are closely linked to the natural Jordan algebra structure on [H.sub.p.sup.d].

We shall now give an example of random quadratic form.

Example 2.1.Let Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]), W [member of] [H.sub.p.sup.d] and [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]), [psi](t) = W [cross product] t. Then (t, [Q.sub.[psi]](Y)) = (Y, [psi](t)(Y)) = (Y, WYt*) = (Y*W*Y, t*) = (Y*WY, t).

Thus [Q.sub.[psi]](Y) = Y* WY. Further, in the case W = [I.sub.n] and [[SIGMA].sub.Y] = [I.sub.n] [cross product] [SIGMA], we have [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d] (n, [SIGMA]) and by (2.3) and Lemma 2.1, (t, E([Q.sub.[psi]](Y))) = 1/d([I.sub.n] [cross product] [SIGMA], [I.sub.n] [cross product] t) = 1/d nd([SIGMA], t). Hence

E([Q.sub.[psi]](Y)) = n[SIGMA]. (2.4)

3. Jordan Algebras

In the following subsections, we shall summarize the notions and results from the theory of Jordan algebras that we require. Only some results are proven. Other results are known and can be found in Braun and Koecher (1), Faraut and Koranyi (4) or Jacobson (6).

The key result in this section is Theorem 3.5.2 which connects the notion of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution.

3.1 Basic Definitions

A Jordan algebra V over the set R of real numbers is a real vector space with a product ab such that ab = ba, [lambda](ab) = ([lambda]a)b, ([a.sub.1] +[a.sub.2])b = [a.sub.1]b+[a.sub.2]b and a([a.sup.2]b) = [a.sup.2](ab) for [lambda] in R and a, [a.sub.1], [a.sub.2] and b in V. An element e in V will be called an identity if ex = x for all x in V.

Let R[X] denote the algebra over R of polynomials in one variable with coefficients in R. For an element x in V, we define R[x] = {p(x): p(X) [member of] R[X]}. The product ab is, in general, nonassociative. In V, define [x.sup.n] recursively by [x.sup.n] = x[x.sup.[n-1]]. V, as Jordan algebra, is power associative, that is [x.sup.p+q] = [x.sup.p][x.sup.q] for all x in V and positive integers p, q. It can be shown that for any p(x) and q(x) in R[x], and a [member of] V, p(x)(q(x)a) = q(x)(p(x)a). An element x in V is said to be invertible with inverse y (denoted [x.sup.-1]) if xy = e and y belongs to R[x]. An element c in V is an idempotent if [c.sup.2] = c. Two idempotents c and d are called orthogonal if cd = 0. An idempotent said to be primitive if it is non-zero and cannot be written as the sum of two non-zero idempotents. A set of idempotents {[c.sub.1], [c.sub.2], ..., [c.sub.m]} in V is called complete if [c.sub.i][c.sub.j] = 0 for i [not equal to] j and [[SIGMA].sub.1.sup.m] [c.sub.i] = e. If, in addition, each [c.sub.i] in this set is primitive the set is called a Jordan frame. A mapping [phi] from V to a Jordan algebra (W, #) is a homomorphism if [phi] is linear and [phi](ab) = [phi](a) # [phi](b) for all a, b in V. Since the polarization identity xy = [[(x + y).sup.2] - [(x - y).sup.2]]/4 holds in V, the linear map [phi] will be a homomorphism iff [phi]([a.sup.2]) = [phi](a) # [phi](a) for all a in V. If V possesses an identity e, [phi](e) will be an identity for [I.sub.m] [phi](the image space of [phi]) but [phi] need not be an identity for W. If [phi] is one to one and onto W, then [phi] is called an isomorphism of V onto W and V and W are said to be isomorphic. A subset I of V is an ideal in V if I is a linear subspace of V and for any x in I, y in V, both xy and yx belong to I; V is said to be simple if its only ideals are {0} and V itself.

It can be shown (see Jacobson (6)) that there exists a unique integer

r > 0 and unique functions [a.sub.j]: V [right arrow] R such that the [a.sub.j] are homogeneous of degree j and for all x in V,

[x.sup.r] - [a.sub.1](x)[x.sup.r-1] + [a.sub.2](x)[x.sup.r-2] - ... + [(-1).sup.r][a.sub.r](x) = 0.

The polynomial

[m.sub.x](X) = [X.sup.r] - [a.sub.1](x)[X.sup.r-1] + [a.sub.2](x)[X.sup.r-2] - ... + [(-1).sup.r][a.sub.r](x)

is called the generic minimum polynomial for x; the degree 'r' of [m.sub.x](X) is called the rank of the Jordan algebra V. The generic trace and generic determinant of x in V are defined by

tr(x) = [a.sub.1](x) and det(x) = [a.sub.r](x).

We shall use upper case notation, "Det", "Tr" to denote the usual trace and determinant for matrices (endomorphisms) and lower case notation "det", "tr" to denote the generic trace and determinant of an element in a Jordan algebra. When required, the notation "[tr.sub.W]"and "[det.sub.W]" will be used to denote the generic trace and determinant with respect to a specific Jordan algebra W.

The space [H.sub.r.sup.1] of Hermitian (symmetric) r x r matrices over R is a Jordan algebra when endowed with the Jordan product A [omicron] B = [AB + BA]/2, with the product on the right side being the usual matrix product. In this case, the generic trace and determinant correspond to the usual trace and determinant for matrices. The notion of Jordan algebra is motivated from this special case.

3.2 Euclidean Jordan Algebras

We say that V is Euclidean if there exists an inner product (., .) on V which is associative. That is (ab, c) = (b, ac) for all a, b and c in V. In every Euclidean Jordan algebra with identity, the generic trace form, (x, y) [right arrow] tr(xy) is positive definite and associative. Unless otherwise stated, we assume that the inner product in a finite dimensional Euclidean Jordan algebra V with identity is given by (x, y) = tr(xy), x, y [member of] V.

For x in V the linear map L(x): V [right arrow] V is defined by L(x)(v) = xv. (Again, for simplicity, L(x)(v) may be written as Lxv. We shall not repeat such convention further.) Further, we define P(x) = 2L[(x).sup.2] - L([x.sup.2]) and P(x, y) = L(x)L(y)+L(y)L(x)-L(xy). Since the inner product is associative, L(x), P(x) and P(x, y) are self-adjoint. The map x [right arrow] P(x) is called the quadratic representation of V. In the Jordan algebra [H.sub.r.sup.1] (with product A [omicron] B = [AB + BA]/2), we have

P(A)B = ABA, P(A,B)C = [ACB+BCA]/2 and P(ABA) = P(A)P(B)P(A).

An element x in V is said to be positive definite (nonnegative definite) if L(x) is positive definite (nonnegative definite). The symmetric cone associated with V is the interior of the set Q = {[x.sup.2]: x [member of] V} and is denoted by [OMEGA](V). It is known that [OMEGA](V) = {x [member of] V: x is positive definite} and [bar.[OMEGA]](V) = Q = {x [member of] V: x is nonnegative definite}.

The following results are well known.

1. For any x in V, x is invertible iff P(x) is invertible whence P[(x).sup.-1] = P([x.sup.-1]) (3.1)

2. For any x, y in V, tr(P(x)y) = tr([x.sup.2]y). (3.2)

3. For any x, y, z in V, P(P(z)x, P(z)y) = P(z)P(x, y)P(z). (3.3)

4. For any x in V and p(x) and q(x) in R[x], P(p(x)q(x)) = P(p(x))P(q(x)) (3.4)

3.3. Pierce and Spectral Decompositions

For an idempotent c in V, the Pierce spaces V(c, i) are defined by

V (c, i) = {x [member of] V: cx = ix}, for i = 0, 1/2, 1.

It is well known that

V = V(c, 1) [direct sum] V(c, 1/2) [direct sum] V(c, 0) (a vector space direct sum)

This decomposition is orthogonal with respect to any associative inner product on V. Also, V(c, 1) and V(c, 0) are Jordan subalgebras of V and c is an identity for V(c, 1). The projections in the Pierce decompositions onto V(c, 1), V(c, 1/2) and V(c, 0) are P(c), I-P(c)-P(e-c) and P(e-c) respectively.

For an illustration of these concepts, take V = [H.sub.r.sup.1], r = p + q and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If V is simple, the value d = dim[V(a, 1/2) [intersection] V(b, 1/2)] is invariant for any pair of orthogonal primitive idempotents (a, b). The value d is called the Pierce invariant and it is related to the dimension and rank of V by n = r+r(r-1)d/2. Moreover, when V is simple, so is V(c, 1) and if c is not primitive, the Pierce invariant for V(c, 1) is also equal to d.

Suppose that rank (V) = r. Then for each x in V, there exists a Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} and [[lambda].sub.i] [member of] R such that

x = [[lambda].sub.1][c.sub.1] + [[lambda].sub.2][c.sub.2] + ... + [[lambda].sub.r][c.sub.r]. (3.5)

The numbers [[lambda].sub.i] (with their multiplicities) are uniquely determined by x and are called the eigenvalues of x. Further, tr(x) = [SIGMA][[lambda].sub.i] and det(x) = [PI][[lambda].sub.i]. The decomposition (3.5) is called the spectral decomposition of x. The rank of x, rk(x), is the number of non-zero eigenvalues (with multiplicities counted) in its spectral decomposition.

For any real-valued function f continuous on a closed interval containing {[[lambda].sub.i]: i = 1, 2, ... r}, define

[~.f](x) = [r.summation over (i = 1)]f([[lambda].sub.i])[c.sub.i]. (3.6)

To check that [~.f](x) is well-defined, i.e., is independent of the Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} chosen for x, let p(X) be a polynomial in R[X]. Then it is straitforward to prove that

[~.p](x) = p(x), (3.7)

which is independent of the chosen Jordan frame for x. So by the Weierstrass approximation theorem, [~.f](x) in (3.6) is determined by f and x. For convenience, we shall write f(x) instead of [~.f](x).

Let

[x.sup.+] = [summation over ([[lambda].sub.i] [not equal to] 0)][[lambda].sub.i.sup.-1]][c.sub.i], [x.sup.0] = [summation over ([[lambda].sub.i] [not equal to] 0)][c.sub.i] and [x.sup.[alpha]] = [summation over ([[lambda].sub.i] [not equal to] 0)][[lambda].sub.i.sup.[alpha]][c.sub.i] (3.8)

where [alpha] may be any real number if all [[lambda].sub.i] are positive and [alpha] may be an integer if some [[lambda].sub.i] are negative. Then [x.sup.+], [x.sup.0], [x.sup.[alpha]] are also well-defined since they may be expressed as f(x) where f is continuous in a closed interval containing the eigenvalues of x: suppose that x = [n.summation over (i=1)] [[lambda].sub.i][c.sub.i]. For simplicity say 0 < [[lambda].sub.1] [less than or equal to] [[lambda].sub.2], ... [less than or equal to] [[lambda].sub.k] with all other [[lambda].sub.i] = 0. Let f(t) = [[lambda].sub.1.sup.[[alpha]-1]]t for 0 [less than or equal to] [[lambda].sub.1] and f(t) = [t.sup.[alpha]] for t > [[lambda].sub.1]. Then f is continuous on [0, [[lambda].sub.k]] with f(0) = 0 and so f(x) = [x.sup.[alpha]] is well-defined. The case where some [[lambda].sub.i] are negative and [alpha] is an integer is similar.

Our motivation for (3.8) arises from the notion of Moore-Penrose inverse in linear algebra and the consideration of Wishart distributions with singular scale matrix.

The following lemmas will be needed later on.

Lemma 3.3.1. Let V be a Euclidean Jordan algebra of rank r with identity e. Then for x [member of] V and functions f and g such that f(x) and g(x) are continuous on a closed interval containing the eigenvalues of x, P(f(x))P(g(x)) = P(f(x)g(x)). In particular, for any u in [bar.[OMEGA]](V), P([u.sup.[alpha]])P([u.sup.[beta]]) = P([u.sup.[alpha]+[beta]]), [alpha], [beta] [member of] R.

Proof. This follows from (3.4) and the Weierstrass approximation theorem.

Lemma 3.3.2. Let V be a Euclidean Jordan algebra of rank r with identity e. (a) Let c [epsilon] V be an idempotent. Then for all x [epsilon] V (c, 1),

[tr.sub.c](x) = [tr.sub.v](x) and [det.sub.c](x) = [det.sub.v](e-c + x).

where [tr.sub.c] and [det.sub.c] are the generic trace and determinant for V (c, 1).

(b) Suppose that the set {[c.sub.1], [c.sub.2], ..., [c.sub.s]} is a family of non-zero orthogonal idempotents (not necessarily primitive) in V and x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i], [[lambda].sub.i] [epsilon] R. Then

[det.sub.V](e-x) = [[product].sub.1.sup.s][(1-[[lambda].sub.i]).sup.tr([c.sub.i])].

Proof. (a) Let x [epsilon] V (c, 1) and suppose that x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i] is a spectral decomposition of x in the Jordan algebra V (c, 1). Then c = [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] since c is the identity for V (c, 1). Extend {[c.sub.1], [c.sub.2], ..., [c.sub.s]} to a Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.s], [c.sub.s+1], ..., [c.sub.r]} for V. Then it is clear that [tr.sub.c](x) = [tr.sub.V] (x). Since e - c + x = [[SIGMA].sup.s.sub.[i=1]] [[lambda].sub.i][c.sub.i] + [[SIGMA].sup.r.sub.[i=1]] [c.sub.i] we have, [det.sub.c](x) = [det.sub.V](e - c + x).

(b) Let [c.sub.s+1] = e - [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] (if [[SIGMA].sup.s.sub.[i=1]] [c.sub.i] = e, s = r and there is nothing to prove). Then {[c.sub.1], [c.sub.2], ..., [c.sub.s+1] is a complete system of orthogonal idempotents. For each i = 1, 2, ..., s+1, let [c.sub.i] = [[SIGMA].sup.ni.sub.[j=1]] [d.sub.ij] be a spectral decomposition on [c.sub.i]. Then [n.sub.i] = tr([c.sub.i]) and the set {[d.sub.ij]: i = 1,2, ..., s+1, j = 1,2, ..., [n.sub.i]} is a Jordan frame for V. Hence

e - x = [[SIGMA].sup.s.sub.[i=1]] [[SIGMA].sup.ni.sub.[j=1]](1-[[lambda].sub.i])[d.sub.ij] + [c.sub.s+1] from which (a) follow.

3.4. Representations, Simple Euclidean Algebras

Let V be a Jordan algebra over R and E a vector space over R. A representation of V on E is a linear map [tau]: V [right arrow] End(E) such that

[tau](xy) = [1/2]([tau](x)[tau](y) + [tau](y)[tau](x)),

i.e. the map [tau] is a Jordan algebra homomorphism of V into End(E) equipped with the Jordan product A [omicron] B = 1/2 (AB + BA). The representation [tau] is said to be self-adjoint if for any x [epsilon] V, [tau] (x) is a self-adjoint endomorphism of E.

It can be proved that there are only four types of simple Euclidean Jordan algebras that admit a self-adjoint representation, namely the spaces [H.sub.r.sup.d], d = 1, 2 and 4 when endowed with the Jordan product A[omicron]B = 1/2 (AB+BA) and the space R x W, where W is a finite-dimensional inner product space, equipped with the product ([[lambda].sub.1], [w.sub.1]) [omicron] ([[lambda].sub.2], [w.sub.2]) = ([[lambda].sub.1][[lambda].sub.2] + ([w.sub.1], [w.sub.2]), [[lambda].sub.1][w.sub.2] + [[lambda].sub.2][w.sub.1]). The Euclidean Jordan algebra [H.sub.3.sup.8] of 3 x 3 Hermitian matrices over the Octonions is simple with Pierce constant d = 8 but does not admit a representation.

If V is a simple Euclidean Jordan algebra of rank r and dimension n, then the following isomorphisms [equivalent] hold.

1. If r = 3, V [equivalent] [H.sub.3.sup.d], d = 1, 2, 4 or 8.

2. If r [greater than or equal to] 4, V [equivalent] [H.sub.r.sup.d], d = 1, 2, or 4.

3. If r = 2, V [equivalent] R x [R.sup.n-1], d = n-2 and in the case that d = 1, 2 or 4(i.e. n = 3,4 or 6), V [equivalent] [H.sub.2.sup.d] [equivalent] R x [R.sup.n-1].

4. If r = 1, V [equivalent] R.

For the Jordan algebras [H.sub.r.sup.d], d = 1, 2, 4, the generic trace and corresponding inner product are given by trA = TrA and (A, B) = tr(A[omicron]B). Note that since A and B are Hermitian, (A, B) = Re Tr(AB) = Re Tr(AB*) so that the inner product for [H.sub.r.sup.d] is simply the inner product inherited from the space [M.sub.[rxr].sup.d] as described in section 2. The generic determinant for [H.sub.r.sup.d] may be obtained from

[(det A).sup.d] = Det [tau](A),

where [tau] is the standard representation for [H.sub.r.sup.d], that is:

1. For A [epsilon] [H.sub.r.sup.1], [tau](A) = A.

2. For A = [A.sub.1] + i[A.sub.2] [epsilon] [H.sub.r.sup.2], [tau] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. For A = [A.sub.1] + i[A.sub.2] + j[A.sub.3] + k[A.sub.4] [epsilon] [H.sub.r.sup.4], [tau] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We need the following lemma.

Lemma 3.4.1. Let V be a simple Euclidean Jordan algebra of rank r [greater than or equal to] 2 with Pierce constant d and identity e. Suppose [tau]: V [right arrow] End(E) is a self-adjoint representation of V on E. Then Tr([tau](e)) is divisible by rd.

Proof. Since [tau](e) is an orthogonal projection, we may choose an orthogonal basis for E such that [tau](e) = diag[[I.sub.k], 0]. Also, since for any x [epsilon] V, [tau](x) = [tau](ex) = 1/2 [[tau](e)[tau](x) + [tau](x)[tau](e)], we must have [tau](x) = diag[[[tau].sub.1](x), 0], where [[tau].sub.1](x) is a k x k matrix. Then [[tau].sub.1] is a self-adjoint representation of V on [R.sup.k] with [[tau].sub.1](e) = [I.sub.k]. The desired result then follows from Theorem 3 and Theorem 6 of Jensen (7).

3.5. Mutations

Let V be a Euclidean Jordan algebra with identity e. Given an element u in V one may define a new composition x * y = P(x, y)u. Then V equipped with the product * is also a Jordan algebra and is called the mutation of V with respect to u and denote by MV (u).

For u [epsilon] V, u[degrees] is an idempotent in V . It will be convenient to use the notation [M.sub.1]V (u) = V (u[degrees], 1) to indicate that the Pierce space V (u[degrees], 1) is being viewed as a subset of the Jordan algebra MV (u). Note that since V (u[degrees], 1) is itself a Jordan algebra containing u, [M.sub.1]V (u) is the mutation of V (u[degrees], 1) with respect to u and hence is a Jordan subalgebra of MV (u). Similarly [M.sub.2]V (u) will denote the orthogonal complement of V (u[degrees], 1) in V viewed as a subset of MV (u). Thus [M.sub.1]V (u) = P(u[degrees])V and [M.sub.2]V (u) = (I - P(u[degrees]))V.

Lemma 3.5.1. Let V be a Euclidean Jordan algebra of rank r with identity e and let u [epsilon] [bar.[OMEGA]](V), u [not equal to] 0, rk(u) = s. Then

(a) MV (u) = [M.sub.1]V (u)[direct sum][M.sub.2]V (u), a vector space direct sum, and for any [alpha] [epsilon] R, [M.sub.1]V (u) = P([u.sup.[alpha]])V = P([u.sup.[alpha]]) V (u[degrees], 1) and ker P([u.sup.[alpha]]) = [M.sub.2]V (u).

(b) The mapping P([u.sup.[1/2]]): MV (u) [right arrow] V is a Jordan algebra homomorphism with ker P([u.sup.[1/2]]) = [M.sub.2]V (u) and ImP([u.sup.[1/2]])=V (u[degrees], 1). Hence [M.sub.2]V (u) is an ideal in MV (u).

(c) The map P([u.sup.[1/2]])|[M.sub.1]V (u) is a Jordan algebra isomorphism from [M.sub.1]V (u) onto V (u[degrees], 1) with inverse P([u.sup.[-1/2]]).

(d) [M.sub.1]V (u) is a Euclidean Jordan algebra of rank s with identity [u.sup.+]. Further for all x [epsilon] [M.sub.1]V (u),

t[r.sub.1](x) = t[r.sub.2](P([u.sup.[1/2]])x) = t[r.sub.V](P([u.sup.[1/2]])x)

and

[det.sub.1](x) = [det.sub.2](P([u.sup.[1/2]])x) = [det.sub.V](e-P([u.sup.[1/2]])([u.sup.+]-x))

where t[r.sub.1], t[r.sub.2], [det.sub.1], [det.sub.2] are the generic traces and determinants for the Jordan algebras M[V.sub.1](u) and V (u[degrees], 1) respectively.

(e) If V is simple, so is [M.sub.1]V (u).

Remark 3.5.1. In this lemma, whenever an operator of the type P(z) (or P(x, y)) is used, it is applied with respect to the product in V, not MV (u).

For example, for x [epsilon] MV (u) we interpret P(z)x to mean 2z(zx) - [z.sup.2]x not 2z * (z * x) - (z * z) * x. Also note that by Lemma 3.3.1, P([u.sup.[alpha]]) P([u.sup.[beta]]) = P ([u.sup.[[alpha]+[beta]]]), [alpha], [beta] [member of] R.

Proof. (a) The fact that MV (u) = [M.sub.1]V (u) [direct sum] [M.sub.2]V (u) is a consequence of the Pierce decomposition of V . Also by (3.7) we have [M.sub.1]V (u) = P(u[degrees])V = P([u.sup.[alpha]]) P([u.sup.-[alpha]])V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] P([u.sup.[alpha]])V = P([u.sup.[alpha]]) P(u[degrees])V = P([u.sup.[alpha]]) V(u[degrees], 1) = P([u.sup.[alpha]]) P(u[degrees])V = P(u[degrees])P([u.sup.[alpha]])V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] P(u.sup.[degrees])V = [M.sub.1]V (u) which proves [M.sub.1]V (u) = P([u.sup.[alpha]])V = P([u.sup.[alpha]])V (u[degrees], 1). Also if x [epsilon] [M.sub.2]V (u), P([u.sup.[alpha]])x= P([u.sup.[alpha]])P(u[degrees])x = 0. Conversely, if P([u.sup.[alpha]])x = 0, then 0 = P([u.sup.-[alpha]])P([u.sup.[alpha]])x = P(u[degrees])x and therefore, x [epsilon] [M.sub.2]V(u).

(b) Let x, y [epsilon] MV (u). Then using (3.3), we have P([u.sup.[1/2]])(x*y) = P([u.sup.[1/2]])P(x, y)u = P([u.sup.[1/2]])P(x, y)P([u.sup.[1/2]])u[degrees] = P(P([u.sup.[1/2]])x, P([u.sup.[1/2]])y)u[degrees] = (P([u.sup.[1/2]])x)(P([u.sup.[1/2]])y). This last equality holds since P([u.sup.[1/2]])x and P([u.sup.[1/2]])y are in V (u[degrees], 1) and u[degrees] is the identity for V (u[degrees], 1). The rest of (b) follows from part (a).

(c) That P([u.sup.[1/2]])|[M.sub.1]V (u) is an isomorphism onto V (u[degrees], 1) follows from part

(b). Further since P([u.sup.[-1/2]]) P([u.sup.[1/2]]) = P(u[degrees]) is the identity map on [M.sub.1]V (u), P([u.sup.[-1/2]]) must be the inverse of P([u.sup.[1/2]])|[M.sub.1]V (u).

(d) Since u[degrees] is the identity for V (u[degrees], 1) and P([u.sup.[-1/2]])u[degrees] = [u.sup.+], part (c) implies [u.sup.+] is the identity for [M.sub.1]V (u). Also by Theorem 1 (vi) p. 224 of Jacobson (6), if [m.sub.x](X) is the generic minimal polynomial for x in [M.sub.1]V (u), then [m.sub.[P([u.sup.[1/2])x]]](X) is the generic minimal polynomial for P([u.sup.[1/2]])x in V (u[degrees], 1). Thus

t[r.sub.1](x) = t[r.sub.2](P([u.sup.[1/2]])x) and [det.sub.1](x) = [det.sub.2](P([u.sup.[1/2]])x).

Further since V (u[degrees], 1) is Euclidean t[r.sub.2] induces an associative inner product on V (u[degrees], 1) and hence t[r.sub.1] does the same for [M.sub.1]V (u). The rest of (d) follows from Lemma 3.3.2(a).

(e) This follows from part (c) and the fact that if V is simple so is V (u[degrees], 1).

The following example will provide a more concrete understanding of Lemma 3.5.1.

Example 3.5.1. Let V = [H.sub.r.sup.1] be the Jordan algebra of r x r Hermitian matrices over R with composition A [omicron] B = 1/2(AB + BA) and let [SIGMA] [epsilon] [H.sub.r.sup.1] be nonnegative definite. Then:

[[SIGMA].sup.+] is the Moore-Penrose inverse of [SIGMA].

[SIGMA][degrees] = [SIGMA][[SIGMA].sup.+], the orthogonal projection of [R.sup.r] onto Im[SIGMA].

[M.sub.1]V ([SIGMA]) = {A [epsilon] V: [SIGMA][degrees]A[SIGMA][degrees] = A}.

[M.sub.2]V ([SIGMA]) = {A [epsilon] V: [SIGMA][degrees]A[SIGMA][degrees] = 0}.

The product in MV ([SIGMA]) is A * B = 1/2 [A[SIGMA]B + B[SIGMA]A] and in [M.sub.1]V ([SIGMA]),

t[r.sub.1](A) = Tr([[SIGMA].sup.[1/2]]A[[SIGMA].sup.[1/2]])

and

[det.sub.1](A) = Det(I-[SIGMA][degrees] + [[SIGMA].sup.[1/2]]A[[SIGMA].sup.[1/2]]).

We are now ready to prove the key result of this section.

Theorem 3.5.2. Suppose that (1) - (5) hold:

(1) J is a Jordan algebra.

(2) J = L [direct sum] K, a vector space direct sum, where

i) L is a Jordan subalgebra of J of rank r with identity [e.sub.L] and L is simple and Euclidean;

ii) K is an ideal in J.

(3) [P.sub.L] and [P.sub.K] are the projections of J onto L and K respectively.

(4) W is a Euclidean Jordan algebra of rank s with identity [e.sub.W].

(5) [rho]: J [right arrow] W is a linear map.

Then (a) - (c) below are equivalent:

(a) [rho] is a Jordan algebra homomorphism with ker[rho] = K.

(b) There exists an integer s > 0 such that for all x [epsilon] J,

det([e.sub.w]-[rho](x)) = [det.sub.L]([e.sub.L]-[P.sub.L]x).sup.s]. (3.10)

(c) There exists an integer s > 0 such that for all x [epsilon] J and k = 1, 2, ...

tr[rho][(x).sup.k] = s t[r.sub.L][([P.sub.L]x).sup.k]. (3.11)

In the case one of (a) - (c) holds s = tr([rho]([e.sub.L]))/r . Also s = tr([rho](c)), where c is any primitive idempotent in the Jordan algebra L.

Proof. (a) [right arrow] (b): Since, by (a), [rho](x) = [rho]([P.sub.L]x) and [P.sub.L]([P.sub.L]x) = [P.sub.L]x, it suffices to show that (3.10) holds for x [epsilon] L.

Let a and b be primitive idempotents in L. By Corollary IV.2.4 of Faraut and Koranyi (4), there exists an element z [epsilon] L such that [z.sup.2] = [e.sub.L] and P(z)a = b. Now by (a), [rho](a) and [rho](b) are non-zero orthogonal idempotents in W. Also, using (3.2), we have rk([rho](b)) = tr([rho](b)) = tr([rho](P(z)a)) = tr(P([rho](z))[rho](a)) = tr([rho][(z).sup.2][rho](a)) = tr([rho]([z.sup.2])[rho](a)) = tr([rho]([e.sub.L])[rho](a)) = tr([rho]([e.sub.L]a)) = tr[rho](a).

Thus for any Jordan frame {[c.sub.1], [c.sub.2], ..., [c.sub.r]} of L, the set {[rho]([c.sub.i]): i = 1, 2, ..., r} is a family of non-zero orthogonal idempotents in W, all of the same rank, say s > 0.

Now let x = [[SIGMA].sup.r.sub.1] [[lambda].sub.j][c.sub.j] be a spectral decomposition of x in L. Then [rho](x) = [[SIGMA].sup.r.sub.1] [[lambda].sub.j][rho]([c.sub.j]) and, by Lemma 3.3.2 (b),

det([e.sub.W] - [rho](x)) = [[product].sub.1.sup.r][(1 - [[lambda].sub.j]).sup.tr([rho]([c.sub.j]))] = [[product].sub.1.sup.r][(1 - [[lambda].sub.j]).sup.s][det.sub.L][([e.sub.L] - x).sup.s].

(b) [right arrow] (c): Fix x [epsilon] J and let [rho](x) = [[SIGMA].sup.r.sub.[j=1]] [[mu].sub.j][d.sub.j] and [P.sub.L]x = [[SIGMA].sup.r.sub.[j=1]] [[lambda].sub.j]([c.sub.j]) be spectral decompositions of [rho](x) in W and [P.sub.L]x in L. Then by (3.10), we have for all [lambda] [not equal to] 0 in R,

det([e.sub.W] - [rho](1/[lambda]x)) = [det.sub.L][([e.sub.L] - [P.sub.L](1/[lambda]x)).sup.s],

or

[[product].sub.1.sup.n]([lambda] - [[mu].sub.j]) = [[lambda].sup.[n-sr]][[product].sub.1.sup.r][([lambda] - [[lambda].sub.j]).sup.s].

Thus tr[rho][(x).sup.k] = [[SIGMA].sup.n.sub.1][[mu].sub.j.sup.k] = s [[SIGMA].sup.r.sub.1][[lambda].sub.j.sup.k] = s [tr.sub.L]([P.sub.L]x).

(c) [right arrow] (a): Let x [epsilon] J. Using the spectral decompositions of [rho](x) and [P.sub.L](x) and using (3.11), it is clear that [rho](x) = 0 iff tr[rho][(x).sup.2] = 0 iff [tr.sub.L][([P.sub.L]x).sup.2] = 0 iff [P.sub.L]x = 0 iff x [epsilon] K. Thus ker [rho] = K.

Further, since K is an ideal, [x.sup.2] = [([P.sub.L]x+[P.sub.K]x).sup.2] = [([P.sub.L]x).sup.2]+z, where z [epsilon] K. Therefore to complete the proof of (a), it suffices to show that [rho]([y.sup.2]) = [rho][(y).sup.2] for y [epsilon] L.

Let y [epsilon] L and let y = [[SIGMA].sup.r.sub.[j=1]] [[lambda].sub.j][c.sub.j] be a spectral decomposition for y in L. By (3.11), tr[rho][([c.sub.j]).sup.k] = s [tr.sub.L][C.sub.j.sup.k] = s, k = 1, 2, ... and so [rho]([c.sub.j]) must be idempotent of rank s in W. Similarly, for i [not equal to] j, [rho]([c.sub.i] + [c.sub.j]) is an idempotent of rank 2s. Then, 2s = tr[rho]([c.sub.i] + [c.sub.j]) = tr([rho][([c.sub.i] + [c.sub.j]).sup.2]) = tr[([rho]([c.sub.i]) + [rho] ([c.sub.j])).sup.2] = tr([rho]([c.sub.i]) + [rho]([c.sub.j]) + 2[rho]([c.sub.i])[rho]([c.sub.j])) = 2s + 2tr([rho]([c.sub.i])[rho]([c.sub.j])). Thus tr([rho]([c.sub.i])[rho]([c.sub.j])) = 0. Hence by problem 7(c) on p. 79 of Faraut and Koranyi (4), [rho]([c.sub.i])[rho]([c.sub.j]) = 0. Therefore

[rho]([y.sup.2]) = [rho]([([r.summation over (j = 1)][[lambda].sub.j][c.sub.j]).sup.2]) = [rho]([r.summation over (j = 1)][[lambda].sub.j.sup.2][c.sub.j]) = [r.summation over (j = 1)][[lambda].sub.j.sup.2][rho]([c.sub.j]) = [([r.summation over (j = 1)][[lambda].sub.j][rho]([c.sub.j])).sup.2] = [rho][(y).sup.2],

proving (a).

Finally, in the case one of (a) - (c) holds, we may take k = 1 and x = [e.sub.L] in (3.11) to obtain s = [tr.sub.[rho]]([e.sub.L])/[tr.sub.L]([e.sub.L]). Also the proof of (a) [right arrow] (b) shows that s = tr([rho](c)) where c is any primitive idempotent in the Jordan algebra L.

Remark 3.5.2. To obtain an example of a Jordan algebra satisfying (1)-(3) of Theorem 3.5.2, consider the Jordan algebra V = [H.sub.r.sup.1] in Example 3.5.1. Then in that example take J = MV([SIGMA]), L = [M.sub.1]V([SIGMA]) and K = [M.sub.2]V([SIGMA]).

Remark 3.5.3. One might wish to investigate, in the context of ideal theory in Jordan algebra, the ideals arisen in Theorem 3.5.2..

4. Main Results. Characterization of the Wishart Distribution

We now characterize the Wishart distribution in terms of Jordan algebra representations (Theorem 4.4). This is accomplished by linking the moment generating function of the Wishart distribution with these homomorphisms via Theorem 3.5.2.

In this section [H.sub.p.sup.d], d = 1, 2, 4, will denote the simple Euclidean Jordan algebras as described in subsection 3.4, M[H.sub.p.sup.d](A) its mutation with respect to an element A [epsilon] [H.sub.p.sup.d] and [H.sub.p.sup.d](c, i), i = 0, 1, 1/2 the Pierce spaces associated with an idempotent c [epsilon] [H.sub.p.sup.d]. Lower case notation 'tr', 'det' refers to the generic trace and determinant and upper case notation 'Tr' 'Det' is the usual trace and determinant for matrices, in this case, for endomorphisms in End ([M.sub.[nxp].sup.d]) or End([R.sup.npd]). We will also make use of the functions [delta] and [phi] as described in Section 2. Also note that End([M.sub.[nxp].sup.d]) is a Euclidean Jordan algebra with identity [I.sub.n] [cross product] [I.sub.p]. Thus for T [epsilon] End([M.sub.[nxp].sup.d]), P(T) is the linear operator given by P(T)S = TST, S [epsilon] End([M.sub.[nxp].sup.d]).

We begin with some results on moment generating functions of quadratic forms.

Lemma 4.1. Let Y ~ [N.sub.k](0, [SIGMA]). Then the moment generating function of YY' is

M(t) = Det[[[I.sub.k] - 2[[SIGMA].sup.[1/2]]t[[SIGMA].sup.[1/2]]].sup.[-1/2]]

for all t [epsilon] [H.sub.k.sup.1] such that [I.sub.k] - 2[[SIGMA].sup.1/2]t[[SIGMA].sup.1/2] is positive definite.

Proof. Let Z ~ [N.sub.k](0, [I.sub.k]). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For t such that [I.sub.k] - 2[[SIGMA].sup.1/2]t[[SIGMA].sup.1/2] is positive definite, a comparison of the integrand with the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] density completes the proof.

Theorem 4.2. (a) Let Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]), [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) be a linear map and [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] the associated quadratic form. Then the moment generating function of [Q.sub.[psi]](Y) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.n] [cross product] [I.sub.P] - 2/d P([[SIGMA].sub.Y.sup.[1/2]])[psi](t) is positive definite.

(b) Let U ~ [W.sub.p.sup.d](m, [SIGMA]), [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then the moment generating function of U is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.p] - 2/d P([[SIGMA].sup.1/2])t [epsilon] [OMEGA]([H.sub.p.sup.d]). Here 'det' is the determinant in the Jordan algebra [H.sub.p.sup.d]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [delta](Y) ~ [N.sub.npd](0, 1/d[phi]([SIGMA]Y)). Lemma 4.1 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.npd] - 2/d[phi][([[SIGMA].sub.Y]).sup.1/2][phi]([psi](t))[phi][([[SIGMA].sub.Y]).sup.1/2] is positive definite. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.n] [cross product] [I.sub.p] - 2/d P([[SIGMA].sub.Y.sup.1/2])[psi](t) is positive definite.

(b) Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where Z ~ [N.sub.[mxp].sup.d] (0, [I.sub.m] [cross product] [SIGMA]) we may apply part (a) with n = m, [[SIGMA].sub.Y] = [I.sub.m] [cross product] [SIGMA] and [psi](t) = [I.sub.m] [cross product] t to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the map x [right arrow] [I.sub.m] [cross product] x is a self-adjoint representation of [H.sub.p.sup.d] on [M.sub.[mxp].sup.d] such that [I.sub.p] [right arrow] [I.sub.m] [cross product] [I.sub.p], we can apply Proposition IV.4.2 of Faraut and Koranyi (4) (with N = mpd and r = p) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [epsilon] [H.sub.p.sup.d] such that [I.sub.p] - 2/d P([[SIGMA].sup.1/2])t [epsilon] [OMEGA]([H.sub.p.sup.d]).

Corollary 4.3. Let Y, [psi], [Q.sub.[psi]] be as in Theorem 4.1, m [epsilon] {1, 2, 3, ...} and [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then (a) - (c) below are equivalent:

(a) [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]).

(b) For all t [epsilon] [H.sub.p.sup.d],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

(c) For all t [epsilon] [H.sub.p.sup.d] and k = 1, 2, ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

Proof. First assume [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]). Then by Theorem 4.2 (a) and (b),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

for all 2/d t [epsilon] [N.sub.0], where [N.sub.0] is a neighbourhood of 0 in [H.sub.p.sup.d]. Now (4.3) amounts to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

for all t [epsilon] [N.sub.0]. Then by analytic continuation, (4.4) holds for all t [epsilon] [H.sub.p.sup.d], proving (b).

Conversely, it is clear that (4.1) implies (4.3), which in turn (by Theorem 4.2) implies [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]).

The equivalence of parts (b) and (c) can be proved by obtaining spectral decompositions of P([[SIGMA].sub.Y.sup.1/2])[psi](t) and P([[SIGMA].sup.1/2])t and using an argument similar to that employed in the proof of (b) [right arrow] (c) in Theorem 3.5.2.

We now prove our main result.

Theorem 4.4. Suppose that:

(1) Y ~ [N.sub.[nxp].sup.d](0, [[SIGMA].sub.Y]).

(2) [psi]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d]) is a linear map.

(3) [Q.sub.[psi]]: [M.sub.[nxp].sup.d] [right arrow] [H.sub.p.sup.d] is the quadratic form associated with the linear map [psi].

(4) [rho]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.[nxp].sup.d] is the linear map defined by [rho](x) = P([[SIGMA].sub.Y.sup.1/2])[psi](x). Then [Q.sub.[psi]](Y) has a Wishart distribution if and only if:

(5) There exists an element [SIGMA] [epsilon][bar.[OMEGA]]([H.sub.p.sup.d]) such that [rho] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.[nxp].sup.d] with ker [rho] = [M.sub.2][H.sub.p.sup.d]([SIGMA]). (When rk([SIGMA]) = 1 the additional condition that Tr[rho]([[SIGMA].sup.+]) is divisible by d is required.) In the case that (5) holds, [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]), where md rk([SIGMA]) = Tr[rho]([[SIGMA].sup.+]). Also md = Tr[rho](c), where c is any primitive idempotent in [M.sub.1][H.sub.p.sup.d]([SIGMA]).

Proof. First assume that [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d](m, [SIGMA]), [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then for x [epsilon] M[H.sub.p.sup.d]([SIGMA]), [[SIGMA].sup.+] - P([SIGMA][degrees])[epsilon] M[H.sub.p.sup.d]([SIGMA]). So by Lemma 3.5.1. (d) and Corollary 4.3, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [det.sub.i], i = 1, 2, 3 are the determinants in [M.sub.1][H.sub.p.sup.d]([SIGMA]), [H.sub.p.sup.d]([SIGMA] [degrees], 1) and [H.sub.p.sup.d] respectively. Then by Theorem 3.5.2 (with J = M[H.sub.p.sup.d]([SIGMA]), L = [M.sub.1][H.sub.p.sup.d]([SIGMA]), K = [M.sub.2][H.sub.p.sup.d]([SIGMA]), W = [End.sub.S]([M.sub.[nxp].sup.d]) and s = md), condition (5) holds.

Conversely, assume (5) holds. Since this is just condition (a) of Theorem 3.5.2 (with J, L, K and W as indicated above), we may apply the equivalent condition (b) of that Theorem together with Lemma 3.5.1 (d) to conclude that there exists an integer s = Tr[rho]([[SIGMA].sup.+])/rk([SIGMA]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [epsilon] M[H.sub.p.sup.d]([SIGMA]). Now by Lemma 3.5.1 (c), [M.sub.1][H.sub.p.sup.d]([SIGMA]) [equivalent] [H.sub.p.sup.d]([SIGMA][degrees] [equivalent] [H.sub.k.sup.d], 1), k = rk([SIGMA]). Thus if rk([SIGMA]) [greater than or equal to] 2, then there exits (by Lemmma 3.4.1) an integer m > 0 such that Tr[rho]([[SIGMA].sup.+]) = md rk([SIGMA]). Hence s = md. So by Corollary 4.3, [Q.sub.[psi]](Y) ~ [W.sub.p.sup.d] (m, [SIGMA]). Also by Theorem 3.5.2, s = md = Tr[rho](c), where c is any primitive idempotent in [M.sub.1][H.sub.p.sup.d]([SIGMA]).

Remarks.

(a) When rk([SIGMA]) = 1, [M.sub.1][H.sub.p.sup.d]([SIGMA]) [equivalent] R for any value of d. This fact is responsible for the requirement that Tr[rho]([[SIGMA].sup.+]) be divisible by d in condition (5).

(b) Equation (2.3) provides a method for selecting the proper [SIGMA] in condition (5).

Corollary 4.5. Let (1) - (4) be as in Theorem 4.3 with [psi](x) = W [cross product] x, W [epsilon] [H.sub.n.sup.d]. Then (a) - (c) below hold:

(a) [Q.sub.[psi]](Y) = Y*WY.

(b) [rho](x) = [[SIGMA].sub.Y.sup.[1/2]] (W [cross product] x)[[SIGMA].sub.Y.sup.[1/2]].

(c) [Q.sub.[psi]](Y) follows a Wishart distribution if and only if there exists an element [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]) such that

(i) ker [rho] = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} and

(ii) for all x [epsilon] [H.sub.p.sup.d],

[[SIGMA].sub.Y.sup.[1/2]](W [cross product] x[SIGMA]x)[[SIGMA].sub.Y.sup.[1/2]] = [[SIGMA].sub.Y.sup.[1/2]](W [cross product] x)[[SIGMA].sub.Y](W[cross product]x)[[SIGMA].sub.Y.sup.[1/2]]. (4.5)

(When rk([SIGMA]) = 1, the additional condition that Tr [rho]([[SIGMA].sup.+]) is divisible by d is required).

In the case (i) and (ii) hold, md rk([SIGMA]) = Tr[rho]([[SIGMA].sup.+]).

Proof. Part (a) was shown in Example 2.1 and part (b) follows from the definition of the operator P([[SIGMA].sub.Y.sup.1/2]). Further, part (c) follows from Theorem 4.4 on noting that by Lemma 3.5.1 (a), [M.sub.2][H.sub.p.sup.d]([SIGMA]) = kerP([SIGMA]) = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} and that equation (4.5) is simply the equation [rho](x * x) = [rho](x[SIGMA]x) = [rho][(x).sup.2].

To illustrate the ideas in Theorem 4.4 and Corollary 4.5, we provide some simple examples.

Example 4.1. In the illustrations below, Y = U + iV ~ [N.sub.2x2.sup.2](0, [[SIGMA].sub.Y]) and [psi]: [H.sub.2.sup.2] [right arrow] [End.sub.S]([M.sub.2x2.sup.2]), [psi](t) = [I.sub.2] [cross product] t. Also, [E.sub.ij] represents the 2 x 2 matrix whose (i, j)th entry is 1 and all other entries 0.

(a) [[SIGMA].sub.Y] = [E.sub.11] [cross product] [E.sub.11].

In this case [Q.sub.[psi]](Y) = Y*Y = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking [rho] as in Corollary 4.5, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Choose [SIGMA] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [epsilon] [H.sub.2.sup.2]. Then [rho]: M[H.sub.2.sup.2]([SIGMA]) [right arrow] End([M.sub.2x2.sup.2) is a self-adjoint Jordan algebra homomorphism with ker [rho] = [M.sub.2][H.sub.2.sup.2]([SIGMA]). Here r = rk([SIGMA]) = 1, d = 2 and Tr([rho]([[SIGMA].sup.+])) = 2. Thus by Corollary 4.5, Y*Y ~ [W.sub.[2x2].sup.2](1, [SIGMA]).

(b) [[SIGMA].sub.Y] is such that for X = A + iB [epsilon] [M.sub.[2x2].sup.2], [[SIGMA].sub.Y] (X) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case, [Q.sub.[psi]](Y) = Y*Y = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for t [epsilon] [H.sub.2.sup.2] the map [rho](t) is given by [rho](t)(A + iB) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking [SIGMA] [epsilon] [H.sub.2.sup.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we see that [rho] is a self-adjoint representation of M[H.sub.2.sup.2]([SIGMA]) on [M.sub.[2x2].sup.2]. (Note that for t [epsilon] [H.sub.2.sup.2], [t.sub.11] [epsilon] R). Here r = rk([SIGMA]) = 1, d = 2 and Tr([rho]([[SIGMA].sup.+])) = 1. Since d does not divide Tr([rho]([[SIGMA].sup.+])), Y*Y is not [W.sub.[2x2].sup.2](m, [SIGMA]). Note however that if we view Y as taking values in [M.sub.2x2.sup.1] and set Z = [square root of (dY)] = [square root of (2Y)], we have Z ~ [N.sub.2x2.sup.1](0, [[SIGMA].sub.Y]) and Z*Z ~ [W.sub.2x2.sup.1](m, [SIGMA]).

Corollary 4.6. Let Y ~ [N.sub.nxp.sup.d](0, A [cross product] [SIGMA]), A [cross product] [SIGMA] [not equal to] 0, W [epsilon] [H.sub.n.sup.d] with AW A [not equal to] 0 and Q(Y) = Y*WY. Then Q(Y) follows a Wishart distribution if and only if

AW AW A = AW A. (4.6)

(When rk([SIGMA]) = 1, the additional condition that Re Tr(AW) be divisible by d is required). In the case (4.6) holds, Q(Y) ~ [W.sub.p.sup.d](m, [SIGMA]), where m = Re Tr(AW).

Proof. First suppose rk([SIGMA]) [greater than or equal to] 2. Let [rho] be as in Corollary 4.5. Then

[rho](x) = [A.sup.1/2]W[A.sup.1/2][cross product][[SIGMA].sup.1/2]x[[SIGMA].sup.1/2].

Since AW A [not equal to] 0, ker [rho] = {x [epsilon] [H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0}. Further, for this particular [rho], (4.6) is equivalent to (4.5). Thus the first part of this result follows from Corollary 4.5. Also if (4.6) holds, Corollary 4.5 and Lemma 2.1 yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus m = Re Tr(AW).

Remark 4.2. Let g be a Hilbert space isomorphism from [M.sub.[nxp].sup.d] onto a Hilbert space F. Then the map

T [right arrow] [T.sub.g] = gT[g.sup.-1]

is an algebra isomorphism from End([M.sub.nxp.sup.d]) onto End(F). Let [psi] and [rho] be as in Theorem 4.4 and define [[psi].sub.g] and [[rho].sub.g] by

[[psi].sub.g](x) = [([psi](x)).sub.g] and [[rho].sub.g](x) = [([rho](x)).sub.g].

Then [[psi].sub.g], [[rho].sub.g]: [H.sub.p.sup.d] [right arrow] [End.sub.S](F) and

[[rho].sub.g](x) = P([([[SIGMA].sub.Y.sup.1/2]).sub.g])[[psi].sub.g](x).

Further, [rho] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.nxp.sup.d] if and only if [[rho].sub.g] is a self-adjoint representation of M[H.sub.p.sup.d]([SIGMA]) on F. Also ker [rho] = ker [[rho].sub.g]. Thus in Theorem 4.4 (Corollary 4.5), it may be easier verify condition (5) (conditions (i) and (ii)) by making a judicious choice for g and using [[psi].sub.g] and [[rho].sub.g] in place of [psi] and [rho].

In particular, if one takes F = [M.sub.pxn.sup.d] and g(X) = X*, X [epsilon] [M.sub.nxp.sup.d], then for all A [epsilon] [M.sub.nxn.sup.d] and B [epsilon] [M.sub.pxp.sup.d],

[(A [cross product] B).sub.g] = B [cross product] A.

Also,

[([[SIGMA].sub.Y]).sub.g] = [[SIGMA].sub.Y*].

Thus in Corollary 4.5, one may replace [rho](x) and [psi](x) by

[[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2] (x [cross product] W)[[SIGMA].sub.Y*.sup.1/2] and [[psi].sub.g](x) = x [cross product] W.

The corollary below is an application of the above principle.

Corollary 4.7. Let Y ~ [N.sub.nxp.sup.d](0, [[SIGMA].sub.Y]). Suppose that [[SIGMA].sub.Y*] = [[SIGMA].sup.r.sub.i=1] [E.sub.ii] [cross product] [A.sub.i], r [less than or equal to] p, where [E.sub.ii] is the p x p matrix whose (i, i)th entry is 1 and all other entries 0, [A.sub.i] [epsilon] [OMEGA]([H.sub.n.sup.d]) and W [epsilon] [H.sub.n.sup.d] with Re Tr([A.sub.i]W) > 0. Then Y* WY follows a Wishart distribution if and only if (1): (1) There exist real numbers [[sigma].sub.k] > 0, k = 1, 2, ..., r such that for all i, j, k,[A.sub.i]W [A.sub.k]W [A.sub.j] = [[sigma].sub.k][A.sub.i]W [A.sub.j]. (When r = 1, the additional condition that 1/[[sigma].sub.1] ReTr([A.sub.i]W) be divisible by d is required.)

In case that (1) holds, Y*WY ~ [W.sub.p.sup.d](m, [SIGMA]), where

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii] and mr = [r.summation over (i = 1)][1/[[sigma].sub.i]]Tr([A.sub.i]W).

Proof. If r = 1, [[SIGMA].sub.Y*] = [E.sub.11] [cross product] [A.sub.1]. So [[SIGMA].sub.Y] = [A.sub.1] [cross product] [E.sub.11]. Letting Z = [square root of ([[sigma].sub.1]Y)] and noting that [[SIGMA].sub.Z] = [A.sub.1] [cross product] [[sigma].sub.1][E.sub.11] and that Y*WY is Wishart if and only if Z*WZ is, the result follows from Corollary 4.6.

Now let r > 1. First assume Y*WY ~ [W.sub.p.sup.d] (m, [SIGMA]). By Corollary 4.5 and Remark 4.2, the map

[[rho].sub.g]: M[H.sub.p.sup.d]([SIGMA])[right arrow]Ends([M.sub.pxn.sup.d]), [[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2](x[cross product]W)[[SIGMA].sub.Y*.sup.1/2]

is a Jordan algebra homomorphism and md rk([[SIGMA].sup.+]) = Tr[[rho].sub.g]([[SIGMA].sup.+]). In this case,

[[rho].sub.g](x) = [[SIGMA].sub.i,j = 1.sup.r] [x.sub.ij][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.j.sup.1/2]. (4.7)

We first determine the nature of [SIGMA]. By Lemma 2.1 with (2.4) and (2.3), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.i] = ([A.sub.i], W) = Re Tr([A.sub.i]W) > 0. Thus

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii], [[sigma].sub.i] = [[a.sub.i]/m]>0.

Now in the Jordan algebra M[H.sub.p.sup.d]([SIGMA]),

x * x = x[SIGMA]x = [summation over (k = 1)][[sigma].sub.k][x.sub.ik][x.sub.kj][E.sub.ij]. (4.8)

Hence by (4.7),

[[rho].sub.g](x * x) = [r.summation over (i,j = 1)][r.summation over (k = 1)][[sigma].sub.k][x.sub.ik][x.sub.kj][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.j.sup.1/2] (4.9)

and

[[rho].sub.g][(x).sup.2] = [r.summation over (i,j = 1)][r.summation over (k = 1)][x.sub.ik][x.sub.kj][E.sub.ij][cross product][A.sub.i.sup.1/2]W[A.sub.k]W[A.sub.j.sup.1/2]. (4.10)

Since [[rho].sub.g](x*x) = [[rho].sub.g][(x).sup.2], a comparison of (4.9) and (4.10) yields [A.sub.i.sup.1/2] W [A.sub.k]W [A.sub.j.sup.1/2] = [[sigma].sub.k][A.sub.i.sup.1/2] W [A.sub.j.sup.1/2] which proves (1).

Conversely, suppose that (1) holds. Define [SIGMA] by

[SIGMA] = [r.summation over (i = 1)][[sigma].sub.i][E.sub.ii].

Then (4.7) - (4.10) hold.

Condition (1) together with (4.9) and (4.10) yields [[rho].sub.g](x * x) = [[rho].sub.g][(x).sup.2].

Also, as Re Tr([A.sub.i]W) > 0, [A.sub.i.sup.1/2]W [A.sub.i.sup.1/2] [not equal to] = 0. Thus by (1), 0 [not equal to] [A.sub.i]W [A.sub.i] = [A.sub.i]W [A.sub.j]W [A.sub.i] and so [A.sub.i]W [A.sub.j] [not equal to] 0. Therefore by (4.7),

ker [[rho].sub.g] = {x[epsilon][H.sub.p.sup.d]: [x.sub.ij] = 0,i,j = 1,2, ... r} = {x[epsilon][H.sub.p.sup.d]: [SIGMA]x[SIGMA] = 0} = [M.sub.2][H.sub.p.sup.d]([SIGMA]).

Thus by Corollary 4.5, Y*WY ~ [W.sub.p.sup.d] (m, [SIGMA]), where md rk([SIGMA]) = Tr[[rho].sub.g]([[SIGMA].sup.+]). (Here rk([SIGMA]) = r > 1). Since [[SIGMA].sup.+] = [[SIGMA].sup.r.sub.i=1] 1/[[sigma].sub.i] [E.sub.ii], [[rho].sub.g]([[SIGMA].sup.+]) = [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] [E.sub.ii] [cross product] [A.sub.i.sup.[1=2]]W [A.sub.i.sup.[1=2]]. Then by Lemma 2.1, Tr[[rho].sub.g]([[SIGMA].sup.+]) = d [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] Re Tr([A.sub.i]W). Hence we have mr = [[SIGMA].sup.r.sub.[i=1]] 1/[[sigma].sub.i] Re Tr([A.sub.i]W).

The following example gives an application of Corollary 4.7 and also provides an illustration of a quadratic form Y*WY that is Wishart but W is not nonnegative definite.

Example 4.2. Let W [epsilon] [H.sub.3.sup.1] and Y ~ [N.sub.[3x2].sup.1] (0, [[SIGMA].sub.Y]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [[SIGMA].sub.Y*] = [[SIGMA].sup.2.sub.i=1] [E.sub.ii] [cross product] [A.sub.i]. So Corollary 4.7 applies. It is easily verified that Tr[A.sub.i]W > 0, i = 1, 2 and that [A.sub.i]W [A.sub.k]W [A.sub.j] = [[sigma].sub.k][A.sub.i]W [A.sub.j], where [[sigma].sub.1] = 4 and [[sigma].sub.2] = 1. Thus Y*WY ~ [W.sub.2.sup.1] (m, [SIGMA]), where m = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Further, letting x = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the Jordan algebra homomorphism [[rho].sub.g](x) = [[SIGMA].sub.Y*.sup.1/2](x [cross product] W) [[SIGMA].sub.Y*.sup.1/2] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now prove a general version of Cochran Theorem (Cochran (2)).

Lemma 4.8. Suppose that:

(1) Y ~ [N.sub.nxp.sup.d] (0, [[SIGMA].sub.Y]);

(2) [[psi].sub.i]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.nxp.sup.d]), i [epsilon] I are linear maps;

(3) [Q.sub.i]:[M.sub.nxp.sup.d] [right arrow] [H.sub.p.sup.d], i [epsilon] I are the quadratic forms associated with the linear maps [[psi].sub.i];

and

(4) [[rho].sub.i]: [H.sub.p.sup.d] [right arrow] [End.sub.S]([M.sub.nxp.sup.d]); i [epsilon] I are the linear maps defined by [[rho].sub.i](x) = P([[SIGMA].sub.Y.sup.[1=2]]) [[psi].sub.i](x).

Then {[Q.sub.i](Y)} is an independent family if and only if

(5) for any distinct i, j [epsilon] I and any u, v [epsilon] [H.sub.p.sup.d], [[rho].sub.i](u)[[rho].sub.j](v) = 0.

Proof. Suppose that {[Q.sub.i](Y)} is an independent family. Let i [not equal to] j and u, v [epsilon] [H.sub.p.sup.d]. Then (u, [Q.sub.i](Y)) = (Y, [[psi].sub.i](u)Y) and (v, [Q.sub.j](Y)) = (Y, [[psi].sub.j](v)Y) are independent. Since

<Y,[[psi].sub.i](u)Y> = <[delta](Y),[phi]([[psi].sub.i](u))[delta](Y)> = [delta](Y)'[phi]([[psi].sub.i](u))[delta](Y)

and [delta](Y) ~ [N.sub.npd](0, 1/d[phi]([[SIGMA].sub.Y])), Theorem 4s p.71 of Searle (18) gives

[1/[d.sup.3]][phi]([[SIGMA].sub.Y])[phi]([[psi].sub.i](u))[phi]([[SIGMA].sub.Y])[phi]([[psi].sub.j](v))[phi]([[SIGMA].sub.Y]) = 0

which is equivalent to condition (5).

Conversely, suppose that (5) holds. We may assume that I = {1, 2 ..., l}. Let Q(Y) = ([Q.sub.i](Y)). To show that {[Q.sub.i](Y)} is an independent family, it suffices to show that [M.sub.Q(Y)](t) = [[product].sup.l.sub.[i=1]] [M.sub.[Q.sub.i](Y)]([t.sub.i]) for t = ([t.sub.i]) in [N.sub.0], where [N.sub.0] is a neighbourhood of 0 in H = [H.sub.p.sup.d] x [H.sub.p.sup.d] x ... x [H.sub.p.sup.d] (l times). Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus by Lemma 4.1,

[[M.sub.Q(Y)](t) = Det [[I.sub.npd] - 2/d [phi]([[[sigma].sub.Y].sup.1/2] ([l.summation over (I=1)] [phi] ([[psi].sub.i]([t.sub.i]))) [phi][([[sigma].sub.Y]).sup.1/2]].sup.1/2] = Det [[[I.sub.n] [cross product] [I.sub.p] - 2/d P [([[SIGMA].sub.Y]).sup.1/2] [l.summation over (I=1)] [[psi].sub.i] [t.sub.i]].sup.-1/2].

Then by condition (5),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 4.9. Let E be a real vector space. Suppose that S, T [epsilon] End(E) are such that [S.sup.2] = S and 1/2 (ST + TS) = T. Then ST = TS.

Proof. Multiplying both sides of 2T = ST + TS on the left by S and then on the right by S yields 2ST = [S.sup.2]T + STS and 2TS = STS + T[S.sup.2]. Since [S.sup.2] = S, ST = STS and TS = STS whence ST = TS.

Lemma 4.10. (a version of Cochran Theorem). Let (1) - (4) be as in Lemma 4.8 and let [SIGMA] [epsilon] [bar.[OMEGA]]([H.sub.p.sup.d]). Then [{[Q.sub.i](Y)}.sub.i[epsilon]I] is an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables for some [m.sub.i] [epsilon] {1, 2, ...} if and only if (a) and (b) below hold. (a) For all i [epsilon] I, [[rho].sub.i] is a self-adjoint representation of the Jordan algebra M[H.sub.p.sup.d]([SIGMA]) on [M.sub.nxp.sup.d] with ker [[rho].sub.i] = [M.sub.2][H.sub.p.sup.d]([SIGMA]) (If rk([[SIGMA].sup.+]) = 1, we also require that Tr[[rho].sub.i]([[SIGMA].sup.+]) is divisible by d.)

(b) For all i [not equal to] j in I, [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.j]([[SIGMA].sup.+]) = 0. In case that (a) and (b) hold, [m.sub.i]d rk([[SIGMA].sup.+]) = Tr [[rho].sub.i]([[SIGMA].sup.+]).

Proof. Let {[Q.sub.i](Y)} be an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables. Then (a) and (b) hold by Theorem 4.4 and Lemma 4.8.

Conversely, suppose that (a) and (b) hold. By Theorem 4.4, [Q.sub.i](Y) ~ [W.sub.p.sup.d] ([m.sub.i], [SIGMA]). Now let u, v [epsilon] [H.sub.p.sup.d]. Then u = [u.sub.1]+[u.sub.2] and v = [v.sub.1]+[v.sub.2], [u.sub.1], [v.sub.1] [epsilon] [M.sub.1][H.sub.p.sup.d]([SIGMA]), [u.sub.2], [v.sub.2] [epsilon] [M.sub.2][H.sub.p.sup.d]([SIGMA]) and for i [not equal to] j in I,

[[rho].sub.i](u)[[rho].sub.j](v) = [[rho].sub.i]([u.sub.1])[[rho].sub.j]([v.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+] * [u.sub.1]) [[rho].sub.j]([[SIGMA].sup.+] * [v.sub.1]). (4.11)

Since [[rho].sub.i]([[SIGMA].sup.+]) = [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.i]([[SIGMA].sup.+]) and [[rho].sub.i]([u.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+] * [u.sub.1]) = [[rho].sub.i]([[SIGMA].sup.+]) [omicron] [[rho].sub.i]([u.sub.1]) = 1/2 [[[rho].sub.i]([[SIGMA].sup.+]) [[rho].sub.i]([u.sub.1])+[[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+])], Lemma 4.9 gives [[rho].sub.i]([[SIGMA].sup.+])[[rho].sub.i]([u.sub.1]) = [[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+]). Similarly [[rho].sub.j]([[SIGMA].sup.+]) [[rho].sub.j]([v.sub.1]) = [[rho].sub.j]([v.sub.1])[[rho].sub.j]([[SIGMA].sup.+]). Thus [[rho].sub.i]([[SIGMA].sup.+]*[u.sub.1])[[rho].sub.j]([[SIGMA].sup.+]*[v.sub.1]) = [[rho].sub.i]([u.sub.1])[[rho].sub.i]([[SIGMA].sup.+]) [[rho].sub.j]([[SIGMA].sup.+])[[rho].sub.j]([v.sub.1]) = 0. Hence by (4.11), [[rho].sub.i](u)[[rho].sub.j](v) = 0 and hence by Lemma 4.8, [[Q.sub.i](Y)} is an independent family.

Corollary 4.11. Let Y ~ [N.sub.nxp.sup.d] (0, A [cross product] [SIGMA]), A [cross product] [SIGMA] [not equal to] 0, [W.sub.i] [epsilon] [H.sub.n.sup.d] with A[W.sub.i]A [not equal to] 0, i [epsilon] I and [Q.sub.i](Y) = Y*[W.sub.i]Y. Then {[Q.sub.i](Y)} is an independent family of [W.sub.p.sup.d] ([m.sub.i], [SIGMA]) random variables if and only if (a) and (b) below hold.

(a) For all i [epsilon] I, A[W.sub.i]A[W.sub.i]A = A[W.sub.i]A. (In the case rk([SIGMA]) = 1, we also require that Re Tr([A.sub.i]W) be divisible by d.)

(b) For all i [not equal to] j in I, A[W.sub.i]A[W.sub.j]A = 0. In case that (a) and (b) hold, [m.sub.i]d rk([SIGMA]) = Tr [[rho].sub.i]([[SIGMA].sup.+]).

Proof. The desired results follow from Theorem 4.10 and Corollary 4.6 on noting that in this case [[psi].sub.i](x) = [W.sub.i] [cross product] x and [[rho].sub.i](x) = [A.sup.1/2][W.sub.i][A.sup.1/2] [cross product] [[SIGMA].sup.1/2]x[[SIGMA].sup.1/2].

Acknowledgement

The authors would like to thank the referee for his careful reading of this paper and for his comments which led to the present improved version.

References

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Joe Masaro [dagger]

Acadia University, Wolfville, Nova Scotia, Canada B4P 2R6

and

Chi Song Wong [double dagger]

University of Windsor, Windsor, Ontario, Canada N9B 3P4

Received June 16, 2008, Accepted Oecember 16, 2008.

* AMS 1991 Mathematics Subject Classification. Primary 62H05; secondary 62H10.

[dagger] E-mail: joe.masaro@acadiau.ca

[double dagger] E-mail: cswong@uwindsor.ca

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Author: | Masaro, Joe; Wong, Chi Song |
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Publication: | Tamsui Oxford Journal of Mathematical Sciences |

Geographic Code: | 9TAIW |

Date: | May 1, 2009 |

Words: | 13774 |

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