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Reduced Triangular Form of Polynomial 3-by-3 Matrices with One Characteristic Root and Its Invariants.

1. Introduction

We consider the following equivalence relation in the set of all polynomial matrices of fixed order over the field C of complex numbers: matrices F(x), G(x) are called semiscalarly equivalent if there exist invertible matrices P, Q(x) over C and C[%], respectively, such that G(x) = PF(x)Q(x) [1] (see also [2]); notation F(x) [approximately equal to] G(x). Several other notions of the equivalence (so-called PS- equivalence) of the polynomial matrices are considered in [3]. Two matrices F(x) and G(x) are said to be PS- equivalent if there exist P(x) [member of] GL(n, C[v]), Q e[member of]GL(n, C) with G(x) = P(x)F(x)Q. If F(x), G(x) are semiscalarly equivalent (or PS- equivalent), then they must have the same characteristic roots and the same invariant factors. By Theorem 1 [1] (see also Theorem 1 [section]1, Section IV [2]) every matrix of full rank is semiscalarly equivalent to the lower triangular form with invariant factors on the main diagonal. The similar results can be found in [4]. However, the matrix of this form is not uniquely defined. Therefore, the question when two matrices are semiscalarly equivalent is open. The conditions of semiscalar equivalence of order 2 polynomial matrices in [5-7] are indicated. In this paper is determined so-called reduced form with respect to semiscalar equivalence for the 3-by-3 matrices with one characteristic root and its invariants are found. The problem of semiscalar equivalence (as of PS- equivalence) contains the classical linear algebra problem of reducing a pair of numerical matrices to a canonical form by a simultaneous similarity transformation (for the solution of this problem, see [8]).

Let F(x) [member of] M(3, C[vj). We assume that characteristic polynomial detF(x) has a unique root. Without loss of generality, we assume that uniquely characteristic root is zero and the first invariant factor of the matrix F(x) is unit. In accordance with [1] at this assumption we have

[mathematical expression not reproducible], (1)

where [mathematical expression not reproducible] (divides). We consider [k.sub.1] < [k.sub.2], since the case [k.sub.1] = [k.sub.2] is considered in [9].

2. Preliminary Results

Proposition 1. In the class {PF(v)Q(v)} of semiscalarly equivalent matrices there exists a matrix of the form (1), in which [mathematical expression not reproducible].

Proof. Proof is obvious.

Let the matrices

[mathematical expression not reproducible], (2)

be given, where [mathematical expression not reproducible].

Proposition 2. A left reducible matrix in the passage from A(x) to the semiscalarly equivalent B(x) of the form (2) is an upper triangular matrix.

Proof. Let A(x) ~ B(x). Then, we have

[mathematical expression not reproducible], (3)

where [[parallel][s.sub.ij][parallel].sup.3.sub.1] [member of] GL(3, C), [[parallel][r.sub.ij](x)[parallel].sup.3.sub.1] [member of] GL(3, C[x]). From (3) it follows that

[mathematical expression not reproducible], (4)

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible]. (6)

Substituting x = 0 in (4), (5), we find that [s.sub.21] = [s.sub.31] = 0. Since [mathematical expression not reproducible], the right-hand side of equality (6) and the second summand of the left-hand side of this equality are divisible by [mathematical expression not reproducible]. Therefore, [mathematical expression not reproducible]. This implies that [s.sub.32] = 0. Proposition is proved.

Proposition 3. If A(x) [approximately equal to] B(x) for the matrices (2), then [mathematical expression not reproducible], where the bracket (, ,) denotes the greatest common divisor.

Proof. Since [r.sub.11] (0) [not equal to] 0 in equality (3), it follows that, from (4) and (5), where [s.sub.21] = [s.sub.31] = [s.sub.32] = 0, we obtain [mathematical expression not reproducible] and [mathematical expression not reproducible], respectively. Thus, [mathematical expression not reproducible]. The notation of semiscalar equivalence is a symmetric relation, so that [mathematical expression not reproducible]. The first part of the Proposition is thus proved. Similarly, from (5) and (6), where [s.sub.33] [not equal to] 0, we can obtain [mathematical expression not reproducible] and [mathematical expression not reproducible], respectively. Therefore, [mathematical expression not reproducible]. Again by virtue of symmetrical relation of semiscalar equivalence we obtain [mathematical expression not reproducible]. The Proposition is proved completely.

Further, by using semiscalarly equivalent transformations A(x) [right arrow] SA(x)R(x) = B(x), we reduce the matrix A(x) to a matrix B(x) of the form (2) with the predefined properties. Furthermore, the left reducible matrix S, obviously, must be selected of the upper triangular form. We shall show how by the given matrix A(x) and by the left reducible matrix S we can find the matrix B(x) of the form (2) and the right reducible matrix R(x) such that A(x) [approximately equal to] B(x) = SA(x)R(x). Then, we shall choose the matrix S of the upper unitriangular form:

[mathematical expression not reproducible]. (7)

By the given entries [a.sub.1](x), [a.sub.2](x), [a.sub.3](x) and [s.sub.12], [s.sub.13], [s.sub.23] of the matrices A(x) and S, respectively, by means of the method of indeterminate coefficients from the congruence

[mathematical expression not reproducible] (8)

we find [b.sub.1] (x) [member of] C[v], deg[b.sub.1] < [k.sub.1]. Denote by [r.sub.uv](x), u, v = 1, 2, such entries:

[mathematical expression not reproducible], (9)

Here, [mathematical expression not reproducible]. Construct the matrix [[parallel][r.sub.uv](x)[parallel].sup.2.sub.1] and consider the congruence

[mathematical expression not reproducible](10)

in the unknowns [b.sub.2](x), [b.sub.3](x). This congruence is solvable, since the free term of the matrix polynomial [[parallel][r.sub.uv](x)[parallel].sup.2.sub.1] is a nonsingular matrix. The unknowns can be found by the method of the indefinite coefficients. It is easily verified that [mathematical expression not reproducible]. Besides the above definition of [r.sub.uv](x), u, v = 1, 2, let us introduce the following notations:

[mathematical expression not reproducible]. (11)

By the indicated above entries [r.sub.ij] (x), i, j = 1, 2, 3, and by the definition from congruence (8), (10) [b.sub.i](x) we construct the matrix [[parallel][r.sub.ij](x)[parallel].sup.3.sub.1] and the matrix B(x) of the form (2). Make sure that the equality SA(x) = B(x)[[parallel][r.sub.ij](x)[parallel].sup.3.sub.1] is valid. This means that the matrix [[parallel][r.sub.ij](x)[parallel].sup.3.sub.1] is invertible and its inverse matrix with S reduces A(x) to B(x). If the matrix S (7) in the passage from A(x) to B(x) has one of the following forms

[mathematical expression not reproducible], (12)

then we shall say that to the matrix A(x) is applied the transformation of type I or the transformation of type II, respectively.

3. Improvement of the Triangular Form of Matrix in the Class of Semiscalarly Equivalent Matrix: Reduced Matrix

Junior degree of polynomial f(x) [member of] C[v], f(x) [not equal to] 0, is the least degree of the monomial (of nonzero coefficient) of this polynomial; notation co deg f. The monomial of degree co deg f and its coefficients are called the junior term and junior coefficients, respectively. Denote by symbol the junior degree of the polynomial f(x) [equivalent to] 0.

Proposition 4. If in the matrix A(x) of the form (2) co deg [a.sub.1] = co deg [a.sub.3] [not equal to] +[infinity], then A(x) [approximately equal to] B(x), where in the matrix B(x) of the form (2) co deg [b.sub.1] > co deg [b.sub.3], co deg [b.sub.2] = co deg [a.sub.2], co deg [b.sub.3] = co deg [a.sub.3].

Proof. We will uniquely determine the value of [s.sub.23] from condition co deg([a.sub.1](x) + [s.sub.23][a.sub.3](x)) > co deg [a.sub.1] and we will apply to the matrix A(x) the transformation of the type I. As a result we obtain the matrix B(x) of the form (2). Its entries [b.sub.i](x), i = 1, 2, 3, satisfy the congruences:

[mathematical expression not reproducible], (13)

[mathematical expression not reproducible], (14)

[mathematical expression not reproducible], (15)

where [mathematical expression not reproducible]. From (14), (15), and (13), we find that co deg [b.sub.2] = co deg [a.sub.2], co deg [b.sub.3] = co deg [a.sub.3] and co deg [b.sub.1] > co deg [b.sub.3], respectively. Proposition is proved.

Proposition 5. Let a matrix A(x) of the form (2) be given such that [mathematical expression not reproducible]. Then there exists a matrix B(x) of the form (2) such that A(x) [approximately equal to] B(x) and co deg [b.sub.1] = co deg [a.sub.1], co deg [b.sub.3] = co deg [a.sub.3], co deg [b'.sub.2] > co deg [b.sub.3], where [mathematical expression not reproducible].

Proof. By transformation of type II we reduce the matrix A(x) to the matrix B(x) of the form (2). Herewith in the matrix S (see (12)) we define [s.sub.12] such that the inequality [mathematical expression not reproducible] is true. The entries [b.sub.1](x), [b.sub.2](x), [b.sub.3](x) of the obtained matrix B(x) satisfy the congruences:

[mathematical expression not reproducible], (16)

[mathematical expression not reproducible], (17)

[mathematical expression not reproducible], (18)

where [mathematical expression not reproducible]. From (16) and (18) we have that co deg [b.sub.1] = co deg [a.sub.1] and co deg [b.sub.3] = co deg [a.sub.3], respectively. If the principle of a choice of [s.sub.12] is considered, then from (17) it follows that co deg [b'.sub.2] > co deg [b.sub.3]. Proposition is proved.

Proposition 6. Let the matrix A(x) have the form (2) and

+[infinity] [not equal to] 2co deg [a.sub.1] + co deg [a'.sub.2] = co deg [a.sub.3] [greater than or equal to] co deg [a.sub.2], (19)

where [mathematical expression not reproducible]. Then there exists a matrix B(x) of the form (2) such that A(x) [approximately equal to] B(x) and [b.sub.1](x) = [a.sub.1](x), co deg [b.sub.2] = co deg [a.sub.2], co deg [b.sub.3] > co deg [a.sub.3].

Proof. Let us apply to the matrix A(x) the transformation of type II. Moreover, in the left reducible matrix (see (12)) we can choose the value of [s.sub.12] so that the condition co deg([a.sub.3] (x) + [s.sub.12][a.sup.2.sub.1](x)[a'.sub.2](x)) > co deg [a.sub.3] is fulfilled. As a result we obtain the matrix B(x) of the form (2) in which its entries [b.sub.i](x), i = 1, 2, 3, satisfy the following congruences:

[mathematical expression not reproducible], (20)

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible]. (22)

From (19) we have that 2co deg [a.sub.1] [greater than or equal to] [k.sub.1]. Then (20) implies [b.sub.1](x) = [a.sub.1](x) and from (22) we find

[mathematical expression not reproducible]. (23)

From (21) and (23) by excluding of [s.sub.12], we arrive at the congruence

[mathematical expression not reproducible], (24)

or

[mathematical expression not reproducible]. (25)

Since [mathematical expression not reproducible], from (21) we have co deg [b.sub.2] = co deg [a.sub.2]. From the inequalities co deg([a.sub.1](x)a'2(x)) < [k.sub.2] - co deg [a.sub.1] and [k.sub.2] - [k.sub.1] + co deg [a.sub.1] [greater than or equal to] [k.sub.2] - co deg [a.sub.1] it follows that junior terms in both members (25) coincide with the junior term of the product [a.sub.1](x)[a'.sub.2] (x). Then from (23), taking into account the choice of [s.sub.12], we find co deg [b.sub.3] > co deg [a.sub.3]. Proposition is proved.

Proposition 7. Let a matrix A(x) of the form (2) be given in which co deg [a.sub.3] [greater than or equal to] co deg [a.sub.2] and 2co deg [a.sub.1] < [k.sub.1]. Then A(x) [approximately equal to] B(x), where B(x) has the form (2) in which co deg [b.sub.2] = co deg [a.sub.2], co deg [b.sub.3] [greater than or equal to] co deg [b.sub.2] and in [b.sub.1] (x) the monomial of degree 2co deg [b.sub.1] = 2co deg [a.sub.1] is absent.

Proof. Let us apply to the matrix A(x) a transformation of type II. In this case, in the left reducible matrix (see (12)) we define [s.sub.12] from the condition [s.sub.12][c.sup.2.sub.0] = [c.sub.1], where [c.sub.0] and [c.sub.1] are junior coefficient and coefficient of the monomial of degree 2co deg [a.sub.1] in [a.sub.1](x), respectively. Then the entries [a.sub.i](x) and [b.sub.i](x) of the matrix A(x) and of the matrix B(x), obtained as a result of transformation, satisfy the congruences (16)-(18). From (16) it follows at once that junior terms in [a.sub.1](x), [b.sub.1] (x) coincide and in [b.sub.1] (x) the monomial of degree 2co deg [a.sub.1] is absent. From (17) and (18) we have, respectively, that co deg [b.sub.2] = co deg [a.sub.2] and co deg [b.sub.3] [greater than or equal to] co deg [b.sub.2]. Proposition is proved.

Taking into account Propositions 4 and 5 we shall think henceforth that co deg [a.sub.3] [not equal to] co deg [a.sub.1] and co deg [a.sub.3] [not equal to] co deg [a'.sub.2] in the matrix A(x) of the form (2), if co deg [a.sub.3] < co deg [a.sub.2]. If co deg [a.sub.3] [greater than or equal to] co deg [a.sub.2], then based on Propositions 6 and 7, we note that in the matrix A(x) the inequality co deg [a.sub.3] [not equal to] 2co deg[a.sub.1] + co deg [a'.sub.2] holds true and in [a.sub.1](x) the monomial of degree 2co deg [a.sub.1] is absent. Moreover, we may take the junior coefficients of the polynomials [a.sub.1](x) and [a.sub.2](x) to be unit, if [a.sub.1](x), [a.sub.1](x) [not equal to] 0. If one of the polynomials [a.sub.1](x), [a.sub.1](x) is identical zero, then we may take the junior coefficients of the nonzero underdiagonal entries of the matrix A(x) to be unit. Such matrix A(x) we shall call the reduced matrix. All subsequent semiscalarly equivalent transformations of the matrix A(x) should not violate her property to be reduced.

4. Invariants of the Reduced Matrix

Theorem 8. In reduced matrix A(x) of the form (2) co deg [a.sub.1], co deg [a.sub.2], and co deg [a.sub.3] are invariants with respect to semiscalarly equivalent transformations.

Proof. Let A(x) and B(x) be reduced matrices of the form (2) and A(x) [approximately equal to] B(x). From equality (3), where matrix [[parallel][s.sub.ij][parallel].sup.3.sub.1] by Proposition 2 is upper triangular, we get

[mathematical expression not reproducible]. (26)

Recall that [s.sub.11], [s.sub.22] [not equal to] 0. If [b.sub.1](x) = 0, then from (26) it follows that [a.sub.1](x) [equivalent to] 0, i.e., co deg [a.sub.1] = co deg [b.sub.1] = +[infinity]. Let [a.sub.1] (x) [not equal to] 0 ([b.sub.1](x) [not equal to] 0). If co deg [a.sub.1] < co deg [a.sub.3], then from (26) at once we have co deg [a.sub.1] = co deg [b.sub.1]. If co deg [a.sub.1] > co deg [a.sub.3], then co deg [a.sub.3] <[k.sub.1] < co deg [a.sub.2]. In view of Proposition 3, we get

[mathematical expression not reproducible]. (27)

Also by Proposition 3 we have [mathematical expression not reproducible]. For this reason co deg [b.sub.1] > co deg [b.sub.3] = co deg [a.sub.3] and from (26) it follows that [s.sub.23] = 0. Thus, co deg [b.sub.1] = co deg [a.sub.1].

From equality (37) we can write

[mathematical expression not reproducible], (28)

[mathematical expression not reproducible]. (29)

We recall that [s.sub.33] [not equal to] 0. If [b.sub.2](x) [equivalent to] 0, then (29) implies that co deg [a.sub.3] = co deg [b.sub.3]. Then, from (28) we find [a.sub.2](x) = 0, since [mathematical expression not reproducible].

Let [a.sub.2](x) [not equal to] 0 ([b.sub.2](x) [not equal to] 0). If co deg [b.sub.3] < co deg [b.sub.2], then (29) implies that co deg [a.sub.3] = co deg [b.sub.3], and from (28), taking into account the form of [r.sub.12] (x) and co deg [a.sub.3] [not equal to] co deg [a'.sub.2], we have co deg [a.sub.2] = co deg [b.sub.2].

If co deg [b.sub.3] [greater than or equal to] co deg [b.sub.2], then from (28) we get also co deg[a.sub.2] = co deg[b.sub.2]. If 2co deg [b.sub.1] < [k.sub.1], then from (26) we obtain [s.sub.12] = 0, [a.sub.1](x) = [b.sub.1](x) and [r.sub.21] (x) = 0. Therefore, from (29) it is clear that co deg [a.sub.3] = co deg [b.sub.3]. In particular, [a.sub.3](x) = 0 [??] [b.sub.3](x) = 0. If 2co deg[b.sub.1] [greater than or equal to] [k.sub.1], then from (26) we have [a.sub.1] (x) = [b.sub.1] (x). In this case, rewrite (29) in the detailed form as

[mathematical expression not reproducible], (30)

where [mathematical expression not reproducible]. Since co deg [[delta].sub.B] = co deg([b.sub.1](x)[b'.sub.2](x)) = co deg([a.sub.1](x)[a'.sub.2](x)) and co deg [a.sub.3], co deg [b.sub.3] [not equal to] 2co deg [a.sub.1] + co deg [a'.sub.2](x), as seen from the last congruence, co deg [a.sub.3] = co deg [b.sub.3]. Moreover, [s.sub.12] = 0, if co deg [a.sub.3] > 2co deg [a.sub.1] + co deg [a'.sub.2] (x). Also in this case from last congruence we obtain [a.sub.3](x) = 0 [??] [b.sub.3](x) = 0. Theorem is proved.

Corollary 9. Let in the reduced matrix A(x) one of the following three conditions hold true:

co deg [a.sub.3] < co deg [a.sub.1] [not equal to] +[infinity], (31)

co deg [a.sub.3] < co deg [a'.sub.2] = [not equal to] +[infinity], (32)

[mathematical expression not reproducible]. (33)

Then left reducible matrix in the passage from A(x) to the reduced matrix B(x) is of the form

[mathematical expression not reproducible], (34)

if condition (31) is fulfilled, or

[mathematical expression not reproducible], (35)

if one from two conditions (32), (33) is valid.

Corollary 10. Identical equality to zero of the entry [a.sub.1] (x), [a.sub.2](x), or [a.sub.3](x) of the reduced matrix A(x) is invariant with respect to semiscalarly equivalent transformations.

Remark. If some two underdiagonal entries in the reduced semiscalarly equivalent matrices A(x), B(x) are nonzero, then diagonal entries of the left reducible matrix, which by Proposition 2 is upper triangular, are equal to each other. Therefore, we can choose this matrix as unitriangular.

Let reduced matrices A(x), B(x) of the form (2) be given. Henceforth we shall apply the following notations:

[mathematical expression not reproducible]. (36)

Corollary 11. If [a.sub.1] (x), [a.sub.2](x) = 0 and co deg[a.sub.3] [greater than or equal to] co deg [a.sub.2] in the reduced matrix A(x), then co deg [[DELTA].sub.A] = co deg([a.sub.1](x)[a.sub.2](x)). Therefore, co deg [[DELTA].sub.A] is an invariant with respect to semiscalarly equivalent transformations.

Theorem 12. In the reduced matrix A(x) the quantity co deg [[DELTA].sub.A] is an invariant with respect to semiscalarly equivalent transformations.

Proof. Let A(x), B(x) be reduced matrices of the form (2) and A(x) [approximately equal to] B(x). Then from equality (3), where matrix [[parallel][s.sub.ij][parallel].sup.3.sub.1] is upper triangular (see Remark), we obtain

[mathematical expression not reproducible], (37)

[mathematical expression not reproducible], (38)

where [mathematical expression not reproducible]. Excluding from (37) and (38) the summand, which contains [s.sub.12], we define

[mathematical expression not reproducible]. (39)

Since [mathematical expression not reproducible], from (39) it follows that co deg [[DELTA].sub.A] = co deg [[DELTA].sub.B].

Corollary 13. The congruence [mathematical expression not reproducible] is an invariant of A(x) with respect to semiscalarly equivalent transformations.

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/3127984

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

References

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[2] P S. Kazimirskii, Factorization of Matrix Polynomials, Naukova Dumka, Kyiv, Ukraine, 1981.

[3] J. A. Dias da Silva and T. J. Laffey, "On simultaneous similarity of matrices and related questions," Linear Algebra and its Applications, vol. 291, no. 1-3, pp. 167-184, 1999.

[4] L. Baratchart, "Un Theoreme de Factorisation et son Application a la Representation des Systemes Cuclique Causaux," Comptes Rendus de l'Academie des Sciences, Series I: Mathematics, vol. 295, no. 3, pp. 223-226, 1982.

[5] B. Z. Shavarovskii, "A complete system of invariants of a second-order matrix with respect to semiscalar equivalence transformations," Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, no. 13, pp. 3-12, 1981.

[6] B. Z. Shavarovskii, "On invariants and canonical form of matrices of second order with respect to semiscalar equivalence," Buletinul Academiei de Stiinfe a Republicii Moldova. Matematica, vol. 82, no. 3, pp. 12-23, 2016.

[7] B. Z. Shavarovskii, "Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices," International Journal of Analysis, vol. 2017, Article ID 6701078, 14 pages, 2017.

[8] S. Friedland, "Simultaneous similarity of matrices," Advances in Mathematics, vol. 50, no. 3, pp. 189-265,1983.

[9] B. Z. Shavarovskii, "Canonical form of polynomial matrices with all identical elementary divisors," Ukrainian Mathematical Journal, vol. 64, no. 2, pp. 282-297, 2012.

B. Z. Shavarovskii (iD)

Department of Algebra, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv 79060, Ukraine

Correspondence should be addressed to B. Z. Shavarovskii; bshavarovskii@gmail.com

Received 21 May 2018; Accepted 26 August 2018; Published 10 September 2018

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