Solutions of the matrix linear bilateral polynomial equation and their structure

We investigate the row and column structure of solutions of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), where A(λ),B(λ) and C(λ) are the matrices over the ring of polynomials F[λ] with coefficients in field F. We establish the bounds for degrees of the rows and columns which depend o...

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Date:	2019
Main Authors:	Dzhaliuk, N.S., Petrychkovych, V.M.
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Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2019
Series:	Algebra and Discrete Mathematics
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Cite this:	Solutions of the matrix linear bilateral polynomial equation and their structure / N.S. Dzhaliuk, V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 243–251. — Бібліогр.: 14 назв. — англ.

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spelling	irk-123456789-1884352023-03-02T01:27:37Z Solutions of the matrix linear bilateral polynomial equation and their structure Dzhaliuk, N.S. Petrychkovych, V.M. We investigate the row and column structure of solutions of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), where A(λ),B(λ) and C(λ) are the matrices over the ring of polynomials F[λ] with coefficients in field F. We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices A(λ) and B(λ). A criterion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices A(λ) and B(λ). 2019 Article Solutions of the matrix linear bilateral polynomial equation and their structure / N.S. Dzhaliuk, V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 243–251. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC: 15A21, 15A24. http://dspace.nbuv.gov.ua/handle/123456789/188435 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We investigate the row and column structure of solutions of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), where A(λ),B(λ) and C(λ) are the matrices over the ring of polynomials F[λ] with coefficients in field F. We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices A(λ) and B(λ). A criterion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices A(λ) and B(λ).
format	Article
author	Dzhaliuk, N.S. Petrychkovych, V.M.
spellingShingle	Dzhaliuk, N.S. Petrychkovych, V.M. Solutions of the matrix linear bilateral polynomial equation and their structure Algebra and Discrete Mathematics
author_facet	Dzhaliuk, N.S. Petrychkovych, V.M.
author_sort	Dzhaliuk, N.S.
title	Solutions of the matrix linear bilateral polynomial equation and their structure
title_short	Solutions of the matrix linear bilateral polynomial equation and their structure
title_full	Solutions of the matrix linear bilateral polynomial equation and their structure
title_fullStr	Solutions of the matrix linear bilateral polynomial equation and their structure
title_full_unstemmed	Solutions of the matrix linear bilateral polynomial equation and their structure
title_sort	solutions of the matrix linear bilateral polynomial equation and their structure
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2019
url	http://dspace.nbuv.gov.ua/handle/123456789/188435
citation_txt	Solutions of the matrix linear bilateral polynomial equation and their structure / N.S. Dzhaliuk, V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 243–251. — Бібліогр.: 14 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT dzhaliukns solutionsofthematrixlinearbilateralpolynomialequationandtheirstructure AT petrychkovychvm solutionsofthematrixlinearbilateralpolynomialequationandtheirstructure
first_indexed	2025-07-16T10:28:23Z
last_indexed	2025-07-16T10:28:23Z
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fulltext	“adm-n2” — 2019/7/14 — 21:27 — page 243 — #93 Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 27 (2019). Number 2, pp. 243–251 c© Journal “Algebra and Discrete Mathematics” Solutions of the matrix linear bilateral polynomial equation and their structure∗ Nataliia S. Dzhaliuk and Vasyl’ M. Petrychkovych Communicated by A. P. Petravchuk Abstract. We investigate the row and column structure of solutions of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), where A(λ), B(λ) and C(λ) are the matrices over the ring of poly- nomials F [λ] with coefficients in field F . We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices A(λ) and B(λ). A cri- terion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices A(λ) and B(λ). Introduction and preliminary results Let F be a field, and F [λ] be a ring of polynomials over F . We denote by M(n,F [λ]) and M(n,m,F [λ]) a ring of n × n matrices and a set of n×m matrices over F [λ], respectively, and by GL(n,F) and GL(n,F [λ]) the groups of invertible n× n matrices over F and F [λ], respectively. ∗This work was supported by the budget program of Ukraine “Support for the development of priority research areas” (CPCEC 6541230). 2010 MSC: 15A21, 15A24. Key words and phrases: matrix polynomial equation, solution, polynomial matrix, semiscalar equivalence. “adm-n2” — 2019/7/14 — 21:27 — page 244 — #94 244 Solutions of the matrix equation We investigate solutions of the matrix linear bilateral polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), (1) where A(λ), B(λ), C(λ) ∈ M(n,F [λ]) are known matrices, and X(λ), Y (λ) ∈ M(n,F [λ]) are unknown ones. Matrix Sylvester-type equations over different domains [2,8], in particular, the matrix polynomial equation (1), appear in various branches of mathematics. Such equations play fundamental role in many problems of control theory and dynamical systems theory [5, 7]. If the matrix polynomial equation (1) is solvable, it obviously has solutions of unbounded above degrees. So, there arises the following natural question: what is minimal degree of solutions of the matrix polynomial equation (1)? This problem was solved only in particular cases. The estimations of degrees of the solutions of the matrix polynomial equation (1) with regular polynomial matrix coefficients, and conditions for uniqueness of such solutions were obtained in [4] and [9]. We recall that collections of polynomial matrices A1(λ), . . . , Ak(λ) and B1(λ), . . . , Bk(λ), where Ai(λ), Bi(λ) ∈ M(n,mi,F [λ]), are called semiscalar equivalent if there exists a scalar matrix Q ∈ GL(n,F) and invertible matrices Ri(λ) ∈ GL(mi,F [λ]) such that Bi(λ) = QAi(λ)Ri(λ), i = 1, . . . , k. It was shown in [6] that the collection of polynomial matrices A1(λ), . . . , Ak(λ) with maximal ranks over algebraically closed field F of characteristic zero is semiscalar equivalent to the collection of the lower triangular polynomial matrices TA1(λ), . . . , TAk(λ) with invariant factors on main diagonals. These results were generalized for polynomial matrices over an arbitrary field F [10,13]. Triangular matrices TAi(λ) are called standard forms of polynomial matrices Ai(λ), i = 1, . . . , k. Note that similar form for one polynomial matrix over infinite field with respect to right semiscalar equivalence of matrices was obtained in [1]. In [2] the solutions of the matrix linear unilateral and bilateral equations over some other rings were investigated. These results were based on the standard form of the pair of matrices with respect to generalized equivalence [11,12]. The conditions for uniqueness of solutions with some properties were also proposed in [2]. In this paper we study solutions of the matrix polynomial equation (1) where both matrix coefficients A(λ) and B(λ) may be nonregular matrices, unlike to [4, 9]. We describe the structure of solutions of this equation “adm-n2” — 2019/7/14 — 21:27 — page 245 — #95 N. S. Dzhaliuk, V. M. Petrychkovych 245 by using standard forms of the collection of polynomial matrices with respect to semiscalar equivalence. We establish the bounds for degrees of solutions of this matrix polynomial equation. A criterion for uniqueness of such solutions and a method for their construction are pointed out. These results are a generalization of results [3]. 1. Structure of solutions of the matrix polynomial equation According to results of [10], a pair of nonsingular polynomial matrices A(λ), B(λ) ∈ M(n,F [λ]) from the matrix polynomial equation (1) is semiscalar equivalent to the pair of triangular matrices TA(λ), TB(λ) in standard form, i.e., there exists an upper unitriangular matrix Q ∈ GL(n,F) and invertible matrices RA(λ) and RB(λ) ∈ GL(n,F [λ]) such that TA(λ) = QA(λ)RA(λ) =   µA 1 (λ) 0 · · · 0 ã21(λ)µ A 1 (λ) µA 2 (λ) · · · 0 · · · · · · · · · · · · ãn1(λ)µ A 1 (λ) ãn2(λ)µ A 2 (λ) · · · µA n (λ)   , (2) where deg ãij(λ) < degµA i (λ)− degµA j (λ) if deg µA i (λ) > degµA j (λ), and ãij(λ) ≡ 0 if µA i (λ) = µA j (λ), i, j = 1, . . . , n− 1, i > j, TB(λ) = QB(λ)RB(λ) =   µB 1 (λ) 0 · · · 0 b̃21(λ)µ B 1 (λ) µB 2 (λ) · · · 0 · · · · · · · · · · · · b̃n1(λ)µ B 1 (λ) b̃n2(λ)µ B 2 (λ) · · · µB n (λ)   , (3) where deg b̃ij(λ) < degµB i (λ)− degµB j (λ) if degµB i (λ) > degµB j (λ), and b̃ij(λ) ≡ 0 if µB i (λ) = µB j (λ), i, j = 1, . . . , n− 1, i > j, µA i (λ) and µB i (λ), i = 1, . . . , n are invariant factors of A(λ) and B(λ), respectively. Triangular matrices TA(λ), TB(λ) can be written in form TA(λ) = T1(λ)S A(λ), TB(λ) = T2(λ)S B(λ), where T1(λ) and T2(λ) are lower unitriangular matrices, SA(λ) and SB(λ) are the Smith normal forms of matrices A(λ) and B(λ). “adm-n2” — 2019/7/14 — 21:27 — page 246 — #96 246 Solutions of the matrix equation Thus, from equation (1) we obtain the following matrix polynomial equation TA(λ)X̃(λ) + Ỹ (λ)TB(λ) = C̃(λ), (4) where X̃(λ) = (RA(λ))−1X(λ)RB(λ), Ỹ (λ) = QY (λ)Q−1, and C̃(λ) = QC(λ)RB(λ). It is easy to verify that the matrix polynomial equation (1) is solvable if and only if the matrix polynomial equation (4) is solvable. Then the following equalities are valid: X(λ) = RA(λ)X̃(λ)(RB(λ))−1 and Y (λ) = Q−1Ỹ (λ)Q (5) for solution X(λ), Y (λ) of (1) and solution X̃(λ), Ỹ (λ) of (4). So, solving of the matrix polynomial equation (1) reduced to solving of the matrix polynomial equation (4). The criterion for solvability of matrix Sylvester-type equations was established by Roth [14]. We denote by rowi(A) and colj(A) the i-th row and j-th column of matrix A, respectively. Further we assume that deg 0 = −∞. Theorem 1. Let SA(λ) = diag(µA 1 (λ), . . . , µ A p (λ), µ A p+1(λ), . . . , µA p+q(λ), µ A p+q+1(λ), . . . , µ A n (λ)), p > 0, q > 0, (6) be the Smith normal form of matrix A(λ) from equation (1), where degµA i (λ) = 0, that is, µA i = 1, i = 1, . . . , p, (7) degµA i (λ) = 1, i = p+ 1, . . . , p+ q, (8) deg µA i (λ) > 1, i = p+ q + 1, . . . , n. (9) If the matrix polynomial equation (4) be solvable, then this equation has solutions in the form X̃1(λ) = [x̃ij(λ)] n i,j=1, Ỹ1(λ) = [ỹij(λ)] n i,j=1 =   Ỹ (p)(λ) Ỹ (q)(λ) Ỹ (n−(p+q))(λ)   , (10) “adm-n2” — 2019/7/14 — 21:27 — page 247 — #97 N. S. Dzhaliuk, V. M. Petrychkovych 247 where Ỹ (p)(λ) =   row1(Ỹ1(λ)) ... rowp(Ỹ1(λ))   , Ỹ (q)(λ) =   rowp+1(Ỹ1(λ)) ... rowp+q(Ỹ1(λ))   , Ỹ (n−(p+q))(λ) =   rowp+q+1(Ỹ1(λ)) ... rown(Ỹ1(λ))   and Ỹ (p)(λ) = 0, (11) Ỹ (q)(λ) is scalar matrix, i.e., Ỹ (q) ∈ M(q, n,F), (12) deg rowi(Ỹ (n−(p+q))(λ)) < deg µA i (λ)− deg(µA i (λ), µ B 1 (λ)), (13) where i = p+ q + 1, . . . , n. Proof. From (4), we obtain the system of linear polynomial equations i∑ l=1 µA l (λ)ãil(λ)x̃lj(λ) + n∑ k=j µB j (λ)̃bkj(λ)ỹik(λ) = c̃ij(λ), (14) where ãii = b̃ii = 1, ãij = b̃ij = 0 if i < j, and C̃(λ) = [c̃ij(λ)] n i,j=1, i, j = 1, . . . , n. It is obvious that the matrix equation (4) has a solution if and only if the system of linear polynomial equations (14) has a solution. Let system (14) be solvable and let x̃ij(λ) = uij(λ), ỹij(λ) = vij(λ), i, j = 1, . . . , n, (15) be solutions of system (14). We split the system (14) into 2n − 1 subsystems in the following way: every t-th subsystem for t 6 n consists of t equations. We obtain it from system (14) by assuming that i = 1, 2, . . . , t and j = n− (t− i), respectively, that is, j = n− (t− 1), n− (t− 2), . . . , n. If t > n, namely t = n+ q, q = 1, 2, . . . , n− 1, then t-th subsystem consists of equations of system (14) for i = q + 1, . . . , n, j = 1, . . . , n− q. The first subsystem of system (14), i.e., t = 1, and therefore i = 1, j = n, consists of one equation: µA 1 (λ)x̃1n(λ) + µB n (λ)ỹ1n(λ) = c̃1n(λ). (16) “adm-n2” — 2019/7/14 — 21:27 — page 248 — #98 248 Solutions of the matrix equation We denote d (A,B) i,j (λ) = (µA i (λ), µ B j (λ)). Since equation (16) is solvable, then d (A,B) 1,n (λ) \| c̃1n(λ). Therefore, we obtain equation: µA 1 (λ) d (A,B) 1,n (λ) x̃1n(λ) + µB n (λ) d (A,B) 1,n (λ) ỹ1n(λ) = c̃1n(λ) d (A,B) 1,n (λ) . (17) Then x̃1n(λ) = u1n(λ), ỹ1n(λ) = v1n(λ) is the solution of (17). We divide v1n(λ) by µA 1 (λ) d (A,B) 1,n (λ) : v1n(λ) = µA 1 (λ) d (A,B) 1,n (λ) s1n(λ)+ ỹ (1) 1n (λ), where deg ỹ (1) 1n (λ) < deg µA 1 (λ) d (A,B) 1,n (λ) . Then x̃ (1) 1n (λ) = u1n(λ) + µB n (λ) d (A,B) 1,n (λ) s1n(λ), ỹ (1) 1n (λ) = v1n(λ)− µA 1 (λ) d (A,B) 1,n (λ) s1n(λ) is the solution of (17) and (16), where deg ỹ (1) 1n (λ) < degµA 1 (λ)− deg(µA 1 (λ), µ B n (λ)). We obtain the second subsystem, that is, t = 2, from (14) by assuming that i = 1, 2 and j = n− 1, n, respectively. This subsystem consists of the following two equations: µA 1 (λ)x̃1,n−1(λ) + µB n−1(λ)ỹ1,n−1(λ) + b̃n,n−1(λ)µ B n−1(λ)ỹ1n(λ) = c̃1,n−1(λ), (18) ãn−1,1(λ)µ A 1 (λ)x̃1n(λ) + µA 2 (λ)x̃2n(λ) + µB n (λ)ỹ2n(λ) = c̃2n(λ). (19) Using the same procedure as in the previous case, we obtain the following solution of equations (18) and (19): x̃ (1) 1,n−1(λ), ỹ (1) 1,n−1(λ), deg ỹ (1) 1,n−1(λ) < degµA 1 (λ)−deg(µA 1 (λ), µ B n−1(λ)), and x̃ (1) 2n (λ), ỹ (1) 2n (λ), deg ỹ (1) 2n (λ) < degµA 2 (λ)− deg(µA 2 (λ), µ B n (λ)). Further, we consider the next subsystem, and so on. Thus, by similar considerations, we obtain solutions x̃ij(λ), ỹij(λ) of system (14) such that deg ỹij(λ) < degµA i (λ) − deg(µA i (λ), µ B 1 (λ)), i, j = 1, . . . , n. From these solutions of system (14) we construct solution X̃(λ), Ỹ (λ) with conditions (11), (12) and (13) of the matrix polynomial equation (4). This completes the proof. “adm-n2” — 2019/7/14 — 21:27 — page 249 — #99 N. S. Dzhaliuk, V. M. Petrychkovych 249 Theorem 2. The solution X̃1(λ), Ỹ1(λ) in form (10) with conditions (11), (12) and (13) of the matrix equation (4) is unique if and only if (µA n (λ), µ B n (λ)) = 1. Proof. It is clear that the matrix polynomial equation (4) has unique solution X̃1(λ), Ỹ1(λ) with conditions (11), (12) and (13) if and only if system (14) has the corresponding unique solution. As in the proof of Theorem 1, the solving of system (14) is reduced to the solving of linear polynomial equations. It is known that linear polynomial equation in form (16) has unique solution with bounded degree deg ỹ (1) 1n (λ) < degµA 1 (λ) − deg(µA 1 (λ), µ B n (λ)) if and only if (µA 1 (λ), µ B n (λ)) = 1. Then system (14) has unique solution with corresponding bounded degree if and only if (µA i (λ), µ B j (λ)) = 1 for all i, j = 1, . . . , n. That is true if and only if (µA n (λ), µ B n (λ)) = 1. Analogously we can describe the column structure of the first compo- nent X̃(λ) of solution X̃(λ), Ỹ (λ) of equation (4). Corollary 1. Let the matrix polynomial equation (4) be solvable. Then it has solutions X̃1(λ) = [x̃ (1) ij (λ)]ni,j=1, Ỹ1(λ) = [ỹ (1) ij (λ)]ni,j=1 such that deg rowi(Ỹ1(λ)) < degµA i (λ)− deg(µA i (λ), µ B 1 (λ)), i = 1, . . . , n, and X̃2(λ) = [x̃ (2) ij (λ)]ni,j=1, Ỹ2(λ) = [ỹ (2) ij (λ)]ni,j=1 such that deg colj(X̃2(λ)) < degµB j (λ)− deg(µA 1 (λ), µ B j (λ)), j = 1, . . . , n. Corollary 2. Let the matrix polynomial equation (1) be solvable. Then it has the following solutions: (i) X1(λ), Y1(λ) such that deg Y1(λ) < degSA(λ)− deg(SA(λ), SB(λ)), (ii) X2(λ), Y2(λ) such that degX2(λ) < degSB(λ)−deg(SA(λ), SB(λ)). Indeed, taking into account the equality (5) between solutions of equations (1) and (4) and a fact that Q is scalar invertible matrix, we obtain solution (i). To obtain solution (ii) we use standard forms T̃A(λ) = R̃A(λ)A(λ)Q̃ and T̃B(λ) = R̃B(λ)B(λ)Q̃ of matrices A(λ) and B(λ) with respect to the right semiscalar equivalence. Theorem 3. If (detA(λ), detB(λ)) = 1 and deg c̃ij(λ) < degµA i (λ) + degµB n (λ), i, j = 1, . . . , n, then the matrix polynomial equation (4) has the solution X̃(λ) = [x̃ij(λ)] n i,j=1, Ỹ (λ) = [ỹij(λ)] n i,j=1 such that deg colj X̃(λ) < degµB n (λ) and deg rowi(Ỹ (λ)) < degµA i (λ), (20) where i, j = 1, . . . , n, and this solution is unique. “adm-n2” — 2019/7/14 — 21:27 — page 250 — #100 250 Solutions of the matrix equation Proof. As in the proof of Theorem 1, we consider the subsystems of system (14). Since (µA 1 (λ), µ B n (λ)) = 1, there exists the following solution of (16): x̃ (1) 1n (λ), ỹ (1) 1n (λ), deg x̃ (1) 1n (λ) < degµB n (λ), deg ỹ (1) 1n (λ) < degµA 1 (λ) and this solution is unique. By substituting ỹ (1) 1n (λ) in (18) instead ỹ1n(λ), we obtain µA 1 (λ)x̃1,n−1(λ) + µB n−1(λ)ỹ1,n−1(λ) = c̃1,n−1(λ)− b̃n,n−1(λ)µ B n−1(λ)ỹ (1) 1n (λ). This equation has solution x̃ (1) 1,n−1(λ), ỹ (1) 1,n−1(λ) such that deg ỹ (1) 1,n−1(λ) < degµA 1 (λ). By comparing the degrees on left and on right parts of the same equation, we obtain deg x̃ (1) 1,n−1(λ) < degµB n (λ). This solution x̃ (1) 1,n−1(λ), ỹ (1) 1,n−1(λ) of equation (18) is also unique, because (µA 1 (λ), µ B n−1(λ)) = 1. In the same way, we obtain unique solution x̃ (1) 2n (λ), ỹ (1) 2n (λ), deg x̃ (1) 2n (λ) < degµB n (λ), deg ỹ (1) 2n (λ) < degµA 2 (λ) from equation (19). Further, we consider the next subsystem, and so on. In this way, we obtain solutions x̃ (1) ij (λ), ỹ (1) ij (λ), i, j = 1, . . . , n, of system (14) and construct solution X̃1(λ) = [x̃ (1) ij (λ)]ni,j=1, Ỹ1(λ) = [ỹ (1) ij (λ)]ni,j=1 of the matrix polynomial equation (4) with conditions (20). This completes the proof. Corollary 3. The matrix polynomial equation (1), where determinants of matrices A(λ) and B(λ) are relatively prime, has solutions: (i) X1(λ), Y1(λ) such that deg Y1(λ) < degSA(λ); (ii) X2(λ), Y2(λ) such that degX2(λ) < degSB(λ). This Corollary is a generalization of results [4, 9] for the matrix poly- nomial equation (1) where the matrix coefficients A(λ) and B(λ) can be nonregular. “adm-n2” — 2019/7/14 — 21:27 — page 251 — #101 N. S. Dzhaliuk, V. M. Petrychkovych 251 References [1] J. A. Dias da Silva, T. J. Laffey, On simultaneous similarity of matrices and related questions, Linear Algebra Appl., 291, 1999, pp. 167-184. [2] N. S. Dzhaliuk, V. M. Petrychkovych, The matrix linear unilateral and bilateral equations with two variables over commutative rings, ISRN Algebra, 2012, Article ID 205478, 14 pages. doi: 10.5402/2012/205478. [3] N. Dzhaliuk, V. Petrychkovych, The semiscalar equivalence of polynomial ma- trices and the solution of the Sylvester matrix polynomial equations, Math. Bull. Shevchenko Sci. Soc., 9, 2012, pp. 81-88. Ukrainian. English summary. [4] J. Feinstein, J. Bar-Ness, On the uniqueness of the minimal solution the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), J. 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Sci., 66 (1), 1993, pp. 2030-2033. [11] V. Petrychkovych, Generalized equivalence of pair of matrices, Linear Multilinear Algebra, 48, 2000, pp. 179-188. [12] V. Petrychkovych, Standard form of pairs of matrices with respect to generalized eguivalence, Visnyk Lviv. Univ., 61, 2003, pp. 153-160. [13] V. Petrychkovych, Generalized Equivalence of Matrices and its Collections and Fac- torization of Matrices over Rings, L’viv: Pidstryhach Inst. Appl. Probl. Mech. and Math. of the NAS of Ukraine, 2015. Ukrainian. English summary. Zbl 1338.15034. [14] W. E. Roth, The equations AX − Y B = C and AX −XB = C in matrices, Proc. Amer. Math. Soc., 3, 1952, pp. 392-396. Contact information N. Dzhaliuk, V. Petrychkovych Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the NAS of Ukraine, Department of Algebra, 3 b, Naukova Str., L’viv, 79060, Ukraine E-Mail(s): nataliya.dzhalyuk@gmail.com, vas_petrych@yahoo.com Web-page(s): www.iapmm.lviv.ua/14/index.htm Received by the editors: 02.07.2018 and in final form 05.12.2018.

Solutions of the matrix linear bilateral polynomial equation and their structure

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