# Right linear map preserving the left spectrum of 2 x 2 quaternion matrices/Kvaternioonide (2 x 2)-maatriksite vasakspektrit sailitavad parempoolsed lineaarsed kujutused.

1. INTRODUCTION

Let R and C be the fields of the real and complex numbers, respectively. The quaternion division ring over R, denoted by H, is the set of all elements with the form [a.sub.0] + [a.sub.1]i + [a.sub.2]j + [a.sub.3]k, where [a.sub.0], [a.sub.1], [a.sub.2], and [a.sub.4] [member of] R; moreover,

[i.sup.2] = [j.sup.2] = [k.sup.2] = i jk = -1;

ij = -ji = k, jk = -kj = i, ki = -ik = j.

Ifa = [a.sub.0] + [a.sub.1]i + [a.sub.2]j + [a.sub.3]k, let

[mathematical expression not reproducible]

be the conjugate, modulus, and real part of a, respectively. It is clear that R [subset] C [subset] H, and the multiplication operation of quaternions is noncommutative.

Let [M.sub.n](R), [M.sub.n](C), and [M.sub.n](H) denote the set of n x n matrices over R, C, and H, respectively Clearly, [M.sub.n](R) [subset] [M.sub.n](C) [subset] [M.sub.n](H). Let E [member of] [M.sub.n](C) denote the identity matrix, [E.sub.ij] [member of] Mn(C) the matrix whose (i,j)th entry is 1 and the other entries are zero. If A [member of] [M.sub.n](C), we write [[sigma].sub.p] (A) as the set of distinct complex eigenvalues of A and [tr.sub.C](A) as the trace of A. In addition, for A = [[a.sub.st]] [member of] [M.sub.n](H), let [A.sup.T] = [[a.sub.ts]] be the transpose of A, observing that there exist unique [A.sub.1], [A.sub.2] [member of] [M.sub.n](C) such that [A.sub.1] = [A.sub.1] +[A.sub.2]j, thus we have [A.sup.T]=[A.sub.1.sup.T]+[A.sub.2.sup.T]j.

Due to the noncommutativity of quaternions, there are two types of eigenvalues and linear maps: left and right eigenvalues of quaternion matrices and left and right quaternion linear maps. This paper only concerns left eigenvalues of a quaternion matrix and right quaternion linear maps, so their definitions are given below, but the introductions to right eigenvalues and left quaternion linear maps are omitted. Readers can refer to [1], [9], and [14] for more information about eigenvalues and linear maps related to the quaternion matrix.

Definition 1.1 ([9,14]). Let A [member of] [M.sub.n](H), [lambda] [member of] His called a left eigenvalue of A if Ax = [lambda]x for some nonzero x [member of] [H.sup.n], where [H.sup.n] is the set of vectors ofn components over H. The set of distinct left eigenvalues is called the left spectrum of A, denoted [[sigma].sub.l](A).

Definition 1.2 ([1,9]). A map [PHI]: [M.sub.n](H) [right arrow] [M.sub.n](H) is said to be a right quaternion linear map if [PHI] satisfies

[PHI](A + B) = [PHI](A) + [PHI](B) and [PHI](Aq) = [PHI](A)q

for all A, B [member of] [M.sub.n](H) and q [member of] H.

As for the studies of the left spectrum of a quaternion matrix, in 1985, Wood [13] used a topological method to show that the left eigenvalue always exists and demonstrated that left eigenvalues of a 2 x 2 quaternion matrix can be found by solving a quaternionic quadratic equation; in 2001, Huang and So [2] computed the left spectrum of a 2 x 2 quaternion matrix by solving quaternionic polynomials of degree 2; in 2005, So [10] also showed that the left spectrum of a 3 x 3 quaternion matrix can be found by this algebraic approach. So far it is still an open problem whether such algebraic approach works for general n x n quaternion matrices for n [greater than or equal to] 4. For other properties and applications about quaternions and quaternion matrices, readers can refer to [9,11,14] and references therein.

Linear preserver problems are the questions about characterizing linear maps on rings or algebras that preserve certain properties, which are a very old and active research area in matrix and operator theory. There has been a great deal of research in this area, especially on spaces of complex matrices. Here, we omit the detailed introduction to linear preserver problems. For some surveys related to the linear preserver problems, readers can consult [3-6,8,11].

From [9,12,14], we can see that there is a 2 x 2 complex matrix whose left spectrum is an infinite set, and the left spectrum of a quaternion matrix is not a similarity invariant in general, thus the classical result ([7, Theorem 3]) about a linear map preserving eigenvalues of complex matrices is not valid for the left spectrum of a quaternion matrix. So we will characterize the form of the linear map preserving the left spectrum of quaternion matrices in this paper.

Considering that the left spectrum of a 2 x 2 quaternion matrix can be found by the explicit formulas introduced in [2], and finding the left spectrum of a quaternion matrix is very difficult to deal with in general, actually so far there is no algorithm for computing the left spectrum of an n x n quaternion matrix for n > 3, so we have decided to work only with 2 x 2 quaternion matrices.

2. PRELIMINARIES

In order to prove the following Theorem 3.2, which characterizes the form of the linear map preserving the left spectrum of 2 x 2 quaternion matrices, we need the following lemmas. For convenience, some known results are also listed as Lemmas 2.1, 2.2, and 2.3.

Lemma 2.1 ([10, Lemma 3.1]). Let A [member of] [M.sub.n](H), and X [member of] [M.sub.n](R) be invertible, then [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]).

Lemma 2.2 ([12, Lemma II.5.1.1 (ii)]). Let A [member of] [M.sub.n](C), then [[sigma].sub.l](A)[intersection]C = [[sigma].sub.p](A).

Lemma 2.3 ([2, Theorem 2.3, Corollary 3.7, Theorem 3.10]). Let A [member of] [M.sub.2](H) and [??].

(1) If bc = 0, then [[sigma].sub.l](A) = {a,d}.

(2) If bc [not equal to] 0, then [[sigma].sub.l](A) = {a + b[lambda]: [[lambda].sup.2] + [b.sup.-1](a-d)[lambda] - [b.sup.-1] c = 0}.

(3) [[sigma].sub.l] (A) is infinite but with a unique modulus and real part if and only if a, b, c, d [member of] R such that [(d-a).sup.2] + 4bc < 0.

(4) Furthermore, if A [member of] [M.sub.2](C), then [[sigma].sub.l](A) is finite if and only if [[sigma].sub.l](A) = [[sigma].sub.p](A).

Remark. The above Lemma 2.3 characterizes the left spectrum of a 2 x 2 quaternion matrix, and will be used many times in the proof of Lemma 2.4. In particular, Lemma 2.3 (3) shows that [[sigma].sub.l] (A) is an infinite set and all elements in [[sigma].sub.l ](A) are of the same modulus and real part if and only if a, b, c, d [member of] R and [(d - a).sup.2] + 4bc < 0. Thus Lemma 2.3 (3) also gives a method for determining whether a 2 x 2 quaternion matrix is a real matrix.

Lemma 2.4. Let [PHI] be a right quaternion linear map from [M.sub.2](H) into itself If [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) for all A [member of] [M.sub.2](H), then [PHI](A)[member of] [M.sub.2](C) for every A [member of] [M.sub.2](C).

Proof. Let

[mathematical expression not reproducible].

Since [[sigma].sub.l]([E.sub.12]) = {0}, by the assumptions of [PHI], one has [[sigma].sub.l]([E.sub.12]) = [[sigma].sub.l]([B.sub.12]) = {0}.

When bc [not equal to] 0, note that [[sigma].sub.l]([B.sub.12]) = {0}, then a + b[lambda] = 0. If a = 0, then b[lambda] = 0. Since bc [not equal to] 0, we have [lambda] = 0. By Lemma 2.3 (2), then [lambda] = 0 satisfies the following equation:

[X.sup.2] + [b.sup.-1](a - d)[lambda]-[b.sup.-1]c = 0.

Consequently, c = 0; this contradicts to bc [not equal to] 0. Hence a [not equal to] 0. By a + b[lambda] = 0, then [lambda] = -[b.sup.-1]a. Again using Lemma 2.3 (2), then

[(-[b.sup.-1]a).sup.2] + [b.sup.-1](a - d)(-[b.sup.-1]a)-[b.sup.-1]c = 0.

Note that a [not equal to] 0, by simple computation, we imply that d[b.sup.-1] = c[a.sup.-1] from the above equality.

Write d[b.sup.-1] = t, then

[mathematical expression not reproducible]

When bc = 0, since [[sigma].sub.l]([B.sub.12]) = {0}, by Lemma 2.3 (1), we have

[B.sub.12] = b[E.sub.12] or [B.sub.12] = c[E.sub.21].

According to the above discussions, then [B.sub.12] has three types of matrix representations. That is,

[mathematical expression not reproducible].

Let

[mathematical expression not reproducible].

Similar to the arguments of [B.sub.12], then [B.sub.21] also has matrix representations

[mathematical expression not reproducible].

In the following, we give the proofs of [B.sub.12] and [B.sub.21] [member of] [M.sub.2](R).

By Lemma 2.3 (3), we imply that [[sigma].sub.l]([E.sub.21] - n[E.sub.12]) is an infinite set and its elements are of the same modulus and real part for n = 1, 2. Note that

[[sigma].sub.l]([E.sub.21] -n[E.sub.12]) = [[sigma].sub.l]([E.sub.21] -n[E.sub.12])) = [[sigma].sub.l][E.sub.21]) -n[PHI]([E.sub.12])) = [[sigma].sub.l]([B.sub.21] -n[B.sub.12]).

Thus [[sigma].sub.l]([B.sub.21] - n[B.sub.12]) is also an infinite set and its elements are of the same modulus and real part for n = 1, 2.

If

apply Lemma 2.3 (3) to [B.sub.21] - [B.sub.12] and [B.sub.21] - 2[B.sub.12], then

a-a', b-b', ta-t'a', tb-t'b' [member of] R.

2a-a', 2b-b', 2ta-t'a', 2tb-t'b' [member of]R.

By simple computation, then [B.sub.12] and [B.sub.21] [member of] [M.sub.2](R).

If

[mathematical expression not reproducible],

also apply Lemma 2.3 (3) to [B.sub.21] -[B.sub.12], we can obtain [B.sub.12] and [B.sub.21] [member of] [M.sub.2](R).

If[B.sub.12] = b[E.sub.12] and

[mathematical expression not reproducible],

applying Lemma 2.3 (3) to [B.sub.21] -[B.sub.12], then [B.sub.12] and [B.sub.21] [member of] [M.sub.2](R).

If [B.sub.12] = b[E.sub.12] and [B.sub.21] = b'[E.sub.12], then [[sigma].sub.l]([B.sub.21] - [B.sub.12]) = {0}. This gives a contradiction because [[sigma].sub.l]([B.sub.21] - [B.sub.12]) is an infinite set. Hence such case is impossible to arise.

If [B.sub.12] = b[E.sub.12] and [B.sub.21] = c'[E.sub.21], apply Lemma 2.3 (3) to [B.sub.21] - [B.sub.12], then b, c' [member of] R. Thus [B.sub.12] and [B.sub.21] [member of] [M.sub.2] (R).

For [??], [B.sub.12] = c[E.sub.21] and [B.sub.21] = b [E.sub.12], as well as [B.sub.12] = c[E.sub.21] and [B.sub.21] = c'[E.sub.21], similar to the above arguments, we can show [B.sub.12] and [B.sub.21] [member of] [M.sub.2](R).

Consequently, we conclude that [PHI]([E.sub.12]) and [PHI]([E.sub.21]) [member of] [M.sub.2](R) from the above proofs.

In the following, we show that [PHI]([E.sub.11]) and [PHI]([E.sub.22]) are also real matrices.

Let

[mathematical expression not reproducible]

Note that [B.sub.12] and [B.sub.21] [member of] [M.sub.2] (R), so we can write [B.sub.11]-[B.sub.12] + [B.sub.21] as

[mathematical expression not reproducible]

where [x.sub.11], [x.sub.12], [x.sub.21], and [x.sub.22] [member of] R.

Apply Lemma 2.3 (3) to [E.sub.11] -[E.sub.12] + [E.sub.21], then [[sigma].sub.l]([E.sub.11] - [E.sub.12] + [E.sub.21]) is an infinite set and its elements are of the same modulus and real part. Since [PHI]([E.sub.11] -[E.sub.12] + [E.sub.21]) = [B.sub.11] -[B.sub.12] +[B.sub.21] and

[[sigma].sub.l]([E.sub.11] - [E.sub.12] + [E.sub.21])) = [[sigma].sub.l]([B.sub.11] - [B.sub.12] + [B.sub.21]),

we also have that [[sigma].sub.l]([B.sub.11] - [B.sub.12] + [B.sub.21]) is an infinite set and its elements are of the same modulus and real part. Again apply Lemma 2.3 (3) to [B.sub.11] -[B.sub.12] + [B.sub.21], then a", b", c", and d" [member of] R. Thus [PHI]([E.sub.11]) [member of] [M.sub.2](R).

Similar to the proof of [PHI]([E.sub.11]) [member of] [M.sub.2] (R), we can show [PHI]([E.sub.22]) [member of] [M.sub.2] (R).

Let A = ([a.sub.ij]) [member of] [M.sub.2](C), note that [PHI] is aright quaternion linear map and [PHI]([E.sub.ij]) [member of] [M.sub.2](R) for i, j = 1,2, then [PHI](A) [member of] [M.sub.2](C). The proof is complete.

By Lemma 2.2 and 2.4, the following lemma 2.5 is valid.

Lemma 2.5. Let [PHI] be a right quaternion linear map from [M.sub.2](H) into itself. If [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) for all A [member of] [M.sub.2](H), then [[sigma].sub.p](A) = [[sigma].sub.p]([PHI](A)) for all A [member of] [M.sub.2](C).

Lemma 2.6. Let X, Y [member of] [M.sub.2](C) wth matrix representations

[mathematical expression not reproducible].

If [[sigma].sub.l](A) = [[sigma].sub.l](XAY) for all A = t[E.sub.12] + s[E.sub.21] [member of] [M.sub.2](C), then

[x.sub.11][y.sub.21], [x.sub.11][y.sub.22], [x.sub.21][y.sub.21], [x.sub.21][y.sub.22], [x.sub.12][y.sub.11], [x.sub.12][y.sub.12], [x.sub.22][y.sub.11], and [x.sub.22][y.sub.21] [member of] R.

Proof. By simple computation, then

[mathematical expression not reproducible].

Take s = 1 and t = -1, then A = [E.sub.21] -[E.sub.12]. By Lemma 2.3 (3), then [[sigma].sub.l](A) is an infinite set and its elements are of the same modulus and real part. Note that the assumption [[sigma].sub.l](A) = [[sigma].sub.l](XAY), and again apply Lemma 2.3 (3) to [[sigma].sub.l](XAY), then

[x.sub.12][y.sub.11] - [x.sub.11][y.sub.21] [member of] R, [x.sub.12][y.sub.12] - [x.sub.11][y.sub.22][member of]R,

[x.sub.22][y.sub.11] -[x.sub.21][y.sub.21] [member of] R, [x.sub.22][y.sub.12] - [x.sub.21][y.sub.22] [member of] R. (1)

Take s = 1 and t = -2, similar to the above arguments, we can show

[x.sub.12][y.sub.11] - 2[x.sub.11][y.sub.21] [member of] R, [x.sub.12][y.sub.12] - 2[x.sub.11][y.sub.22] [member of] R,

[x.sub.22][y.sub.11] - 2[x.sub.21][y.sub.21] [member of] R, [x.sub.22][y.sub.12] - 2[x.sub.21][y.sub.22] [member of] R. (2)

With equalities (1) and (2), by simple computation, we can imply that Lemma 2.6 holds.

Lemma 2.7. Let X [member of] [M.sub.2](C) be invertible. If [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]) for all A = t[E.sub.12] + s[E.sub.21] [member of] [M.sub.2](C) and A = diag(s, t)[member of] [M.sub.2](C), then there exist [theta] [member of] [0,2[pi]) and an invertible matrix B [member of][M.sub.2](R) such that X = [e.sup.i[theta]] B.

Proof. Note that X [member of] [M.sub.2](C) is invertible, let

[mathematical expression not reproducible].

Since A = diag(s,t) [member of] [M.sub.2](C), by simple computation, one has

[mathematical expression not reproducible].

Take s = 1, t = 0, by Lemma 2.3 (1), then [[sigma].sub.l](A) = {0, 1}. Since

Xdiag(1, 0)[X.sup.-1] [member of][M.sub.2](C) and [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]),

by Lemma 2.3 (4), we have [[sigma].sub.p](XA[X.sup.-1]) = {0,1}. Hence, [tr.sub.C](XA[X.sup.-1]) = 1, that is

[x.sub.11][y.sub.11] + [x.sub.21][y.sub.12] = 1. (3)

Take s = 0, t = 1, similar to the above arguments, we have

[x.sub.12][y.sub.21] + [x.sub.22][y.sub.22] = 1. (4)

Since X[X.sup.-1] = E, by simple computation, one has

[x.sub.11][y.sub.11] + [x.sub.12][y.sub.21] = 1and[x.sub.21] [y.sub.12] + [x.sub.22][y.sub.22] = 1. (5)

By equalities (3), (4), and (5), then

[x.sub.12][y.sub.21] = [x.sub.21] [y.sub.12] and [x.sub.11][y.sub.11] = [x.sub.22][y.sub.22]. (6)

Note thatX [member of] [M.sub.2](C), then there exist [[thata].sub.ij] [member of] [0, 2[pi]), i, j = 1,2, such that

[mathematical expression not reproducible] (7)

Case 1. [x.sub.11][x.sub.12][x.sub.21][x.sub.22] [not equal to] 0.

Since [x.sub.11][x.sub.12][x.sub.21][x.sub.22] [not equal to] 0, note that X [member of] [M.sub.2](C) and the assumption [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]) for every A = t[E.sub.12] + s[E.sub.21] [member of] [M.sub.2](C), by Lemma 2.6 and equality (7), we can imply that there exist [a.sub.21], [a'.sub.22] [a.sub.22], [a'.sub.22], [b.sub.11], [b'.sub.11], [b.sub.12], and [b'.sub.12] [member of] R such that

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

By equalities (3) and (4), then [y.sub.11] [not equal to] 0 or [y.sub.12] [not equal to] 0; moreover, [y.sub.21] [not equal to] 0 or [y.sub.22] [not equal to] 0. Using equalities (9) and (8), one has

[[theta].sub.11] = [[theta].sub.21] or [[theta].sub.11][+ or -] [pi] = [[theta].sub.21], (10)

[[theta].sub.12] = [[theta].sub.22] or [[theta].sub.12] [+ or -] [pi] = [[theta].sub.22]. (11)

In addition, by equalities (4), (7), and (9), then

[mathematical expression not reproducible].

Combining the above equality with equality (11), we obtain that

[[theta].sub.11] = [[theta].sub.12] or [[theta].sub.11] [+ or -] [pi] = [[theta].sub.12]. (12)

Let [theta] = [[theta].sub.11], by equalities (10), (11), and (12), then X is one of the following matrices

[mathematical expression not reproducible].

Consequently, there are invertible matrices B [member of] [M.sub.2](R) and [theta] [member of] [0,2[pi]) such that X = [e.sup.i[theta]]B.

Case2.[x.sub.11][x.sub.12][x.sub.21][x.sub.22] = 0.

(a) When [x.sub.11] = 0. By equalities (3) and (5), then

[x.sub.21][y.sub.12] = [x.sub.12][y.sub.21] = 1. (13)

By equalities (13) and (7), then there exist [c.sub.12] and [c.sub.21] [member of] R such that

[mathematical expression not reproducible].

By equality (13), then [x.sub.12] [not equal to] 0, [y.sub.12] [not equal to] 0. In terms of Lemma 2.6, we have [x.sub.12][y.sub.12] [member of] R. So there exists [c'.sub.12] [member of] R such that [??]. Note that [??] and [y.sub.12] [not equal to] 0, then

[[theta].sub.12] = [[theta].sub.21] or [[theta].sub.12] [+ or -] [pi] = [[theta].sub.21]. (14)

Since [x.sub.11] = 0, by equality (6), one has [x.sub.22][y.sub.22] = 0. By Lemma 2.6, we know [x.sub.22][y.sub.21] [member of] R. If [x.sub.22] [not equal to] 0, then there exists [c'.sub.21] [member of] R such that [??]. Since [??] and [y.sub.21] [not equal to] 0, we have

[[theta].sub.12] = [theta].sub.22]or [theta].sub.12] [+ or -] [pi] = [theta].sub.22]. (15)

If [x.sub.22] = 0, write [??]. Consequently, let [theta] = [theta].sub.12], by equalities (7), (14), and (15), then X is one of the following matrices

[mathematical expression not reproducible],

Hence, there are invertible matrices B [member of] [M.sub.2](R) and [theta] [member of] [0,2[pi]) such that X = [e.sup.i[theta]B.

(b) When [x.sub.22] = 0. Similar to the proof of Case (a), we can show that there are invertible matrices B [member of][M.sub.2](R) and 0 [member of] [0,2[pi]) such that X = [e.sup.i[theta]B.

(c) When [x.sub.12] = 0. By equalities (4), (6), and (7), then [x.sub.11][y.sub.11] = [x.sub.22][y.sub.22] = 1. Thus there exist [d.sub.11] and [d.sub.21] [member of] R such that [??] and [??]. By Lemma 2.6, then [x.sub.11][y.sub.22] and [x.sub.21][y.sub.22] [member of] R. Hence there exists[d'.sub.11] [member of] R such that [??]. Since [y.sub.22] [not equal to] 0, one has

[[theta].sub.11] = [[theta].sub.22] or [[theta].sub.11] [+ or -] [pi] = [[theta].sub.22]. (16)

Note that [x.sub.21][y.sub.22] [member of] R, if [x.sub.21] = 0, then there exists [d.sub.21] [member of] R such that [??]. Again using [??] and [y.sub.22] [not equal to] 0, then

[[theta].sub.21] = [[theta].sub.22] or [[theta].sub.21] [+ or -] [pi] = [[theta].sub.22], (17)

if [x.sub.21] = 0, write [??]. Consequently, let [theta] = [[theta].sub.11], by equalities (7), (16), and (17), then X is of one of the following forms:

[mathematical expression not reproducible].

Hence, there are invertible matrices B [member of] [M.sub.2](R) and [theta] [member of] [0,2%) such that X = [e.sup.i[theta]]B.

(d) When[x.sub.21] =0. Analogous to the proof of Case (c), we can also show that there are invertible matrices B [member of] [M.sub.2](R) and 0 [member of] [0,2[pi]) such that X = [e.sup.i[theta]]B.

In conclusion, by the above arguments, the proof is completed.

Lemma 2.8. There exists A [member of] [M.sub.2](H) such that [[sigma].sub.l](A) = [[sigma].sub.l]([A.sup.T]). Proof. Let A be the same as that of [14, Example 7.3], that is

[mathematical expression not reproducible].

If Ax = 0, where x = [([x.sub.1], [x.sub.2]).sup.T], by simple computation, then x = 0. Consequently, 0 [??] [[sigma].sub.l](A). Take y = [(-i, k).sup.T], then [A.sup.T]y = 0, we have 0 [member of] [[sigma].sub.l]([A.sup.T]). Hence [[sigma].sub.l](A) [not equal to] [[sigma].sub.l]([A.sup.T]).

3. RIGHT LINEAR MAP PRESERVING THE LEFT SPECTRUM

With the preparations in Section 2, we prove the following Theorem 3.2. For convenience, we also list [7, Theorem 3] as Lemma 3.1 in the following.

Lemma 3.1 ([7, Theorem 3]). Let [PHI]: [M.sub.n](C) [right arrow] [M.sub.n](C) be a linear map, then [[sigma].sub.p](A) = [[sigma].sub.p]([PHI](A)) for all A [member of] [M.sub.n](C) if and only if there exists an invertible matrix X [member of] [M.sub.n](C) such that

[PHI](A)=XA[X.sup.-1] or [PHI](A)=X[A.sup.T][X.sup.-1].

Theorem 3.2. Let [PHI] be a right quaternion linear map from [M.sub.2](H) into itself Then [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) for all A [member of] [M.sub.2](H) if and only if there exists an invertible matrix B [member of] [M.sub.2](R) such that [PHI](A) = BA[B.sup.-1].

Proof. The sufficiency follows from Lemma 2.1. In the following, we prove the necessity of Theorem 3.2.

If [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) for all A [member of] [M.sub.2](H), by Lemma 2.5, we imply that

[[sigma].sub.p](A) = [[sigma].sub.p]([PHI](A))

for all A [member of] [M.sub.2](C). By Lemma 3.1, then there exists an invertible matrix X [member of] [M.sub.2](C) such that [PHI](A)=XA[X.sup.-1] or [PHI](A)=X[A.sup.T][X.sup.-1]

for all A [member of] [M.sub.2](C).

Case 1. If [PHI](A) =XA[X.sup.-1] for all A [member of] [M.sub.2](C).

Since X [member of] [M.sub.2](C) and [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]) for all A [member of] [M.sub.2](H), by Lemma 2.7, we imply that there exist invertible matrices B [member of] [M.sub.2](R) and [theta] [member of] [0,2[pi]) such that X = [Be.sup.i[theta]]. Note that [PHI] is a right quaternion linear map; moreover, A [member of] [M.sub.2](H) can be uniquely expressed as A = [A.sub.1] + [A.sub.2]j, where [A.sub.1] and [A.sub.2] [member of] [M.sub.2](C), then

[mathematical expression not reproducible] (18)

Note that B is a real matrix, then [B.sup.-1]j = j[B.sup.-1]. Since X = [e.sup.i[theta]] B, [A.sub.1], [A.sub.1] [member of] [M.sub.2](C), by equality (18), we have

[mathematical expression not reproducible].

Case 2. If [PHI](A) = X[A.sup.T][X.sup.-1] for all A [member of] [M.sub.2](C).

Let [X.sup.-1] = Y, then [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) = [[sigma].sub.l]( X[A.sup.T]Y) for all A [member of] [M.sub.2](C). Hence, [[sigma].sub.l](A) = [[sigma].sub.l](X[A.sup.T]Y) for all A = t[E.sub.12] + s[E.sub.21] [member of] [M.sub.2](C) and A = diag(s,t) [member of] [M.sub.2](C). Since t[E.sub.12] + s[E.sub.21] = [(t[E.sub.12] + s[E.sub.21]).sup.T] and diag (s, t ) = diag[(s, t).sup.T], by Lemma 2.6 and 2.7, we obtain that there exist invertible matrices B [member of] [M.sub.2](R) and d [member of] [0,2[pi]) such that X = [e.sup.i[theta]]B. Similar to Case 1, we can show that

[mathematical expression not reproducible] (19)

for all A = [A.sub.1] +[A.sub.2]j, where [A.sub.1] and [A.sub.2] [member of] [M.sub.2](C). Since B is an invertible real matrix, moreover, [[sigma].sub.l](A) = [[sigma].sub.l]([PHI](A)) for all A [member of] [M.sub.2](H), by equality (19) and Lemma 2.1, we have

[[sigma].sub.l](A) = [[sigma].sub.l]([A.sup.T])

for every A [member of] [M.sub.2](H). By Lemma 2.8, a contradiction is yielded. Hence Case 2 is impossible to arise.

By the above arguments, the proof is completed.

4. CONCLUSIONS

We have studied the problem of a right linear map preserving the left spectrum of 2 x 2 quaternion matrices and characterized the form of such map. By Lemma 3.1 and Theorem 3.2, it is easy to see that the form of a right linear map preserving the left spectrum of 2 x 2 quaternion matrices is not analogous to the classical results of the linear map preserving eigenvalues of n x n complex matrices for n [greater than or equal to] 2. By Lemma 2.1 and Theorem 3.2, we conjecture that, for n > 2, the right quaternion linear map [PHI] preserves the left spectrum of n x n quaternion matrices if and only if there exists an invertible n x n real matrix X such that [PHI](A) =XA[X.sup.-1] for every n x n quaternion matrix A. This will be dealt with in future works.

ACKNOWLEDGEMENTS

The authors are very grateful to the referee whose comments have greatly improved the final version of this article. The authors also thank Professor Liping Huang for his explanation to Lemma 2.3 (3). This work was supported in part by the project of the 12th five-year plan of eduction science of the Shandong Province in China (grant No. YBS15014). The publication costs of this article were partially covered by the Estonian Academy of Sciences.

REFERENCES

[1.] Farenick, D. R. and Pidkowich, B. A. F. The spectral theorem in quaternions. Linear Algebra Appl., 2003, 371, 75-102.

[2.] Huang, L. P. and So, W. On left eigenvalues of a quaternionic matrix. Linear Algebra Appl., 2001, 323, 105-116.

[3.] Jafarian, A. A. A survey of invertibility and spectrum preserving linear maps. Bull. Iran. Math. Soc, 2009, 2, 1-10.

[4.] Jafarian, A. A. and Sourour, A. R. Spectrum-preserving linear maps. J. Funct. Anal., 1986, 66, 255-261.

[5.] Li, C. K. and Tsing, N. K. Linear preserver problems: a brief introduction and some special techniques. Linear Algebra Appl., 1992, 162-164, 217-235.

[6.] Marcus, M. Linear transformations on matrices. J. Res. Nat. Bur. Standards Sect. B., 1971, 75B, 107-113.

[7.] Marcus, M. and Moyls, B. N. Linear transformations on algebras of matrices. Canad. J. Math., 1959, 11, 61-66.

[8.] Pierce, S. A survey of linear preserver problems. Linear Multilinear Al., 1992, 33, 1-129.

[9.] Rodman, L. Topics in Quaternion Linear Algebra. Princeton Univ. Press, Princeton, USA, 2014.

[10.] So, W. Qnaternionic left eigenvalue problem. Southeast Asian Bull. Math, 2005, 29, 555-565.

[11.] Sourour, A. R. Three linear perserver problems. In Mathematics and the 21st Century. World Sci. Publ., London, 2001, 211-221.

[12.] Siu, L. S. A Study of Polynomials, Determinants, Eigenvalues and Numerical Ranges Over Quaternions. M. Phil. thesis, University of Hong Kong, 1997.

[13.] Wood, R. M. W. Quaternionic eigenvalues. Bull. London Math. Soc, 1985, 17, 137-138.

[14.] Zhang, F. Z. Quaternions and matrices of quaternions. Linear Algebra Appl., 1997, 251, 21-57.

Deyu Duan (a), Xiang Gonga, Geng Yuan (b), and Fahui Zhai (b*)

(a) Research Center of Mathematical Modeling, Qingdao University of Science and Technology, Qingdao 266061, China

(b) Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China

(*) Corresponding author, fahuiz@163.com

Received 12 March 2018, accepted 9 July 2018, available online 23 October 2018

https://doi.org/10.3176/proc.2018.4.08