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ISRN Algebra
Volumeย 2012ย (2012), Article IDย 205478, 14 pages
http://dx.doi.org/10.5402/2012/205478
Research Article

The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3-b, Naukova Street, 79060 L'viv, Ukraine

Received 16 January 2012; Accepted 20 February 2012

Academic Editors: I.ย Cangul, H.ย Chen, and P.ย Damianou

Copyright ยฉ 2012 N. S. Dzhaliuk and V. M. Petrychkovych. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The method of solving matrix linear equations ๐ด๐‘‹+๐ต๐‘Œ=๐ถ and ๐ด๐‘‹+๐‘Œ๐ต=๐ถ over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is proposed. The formulas of general solutions of such equations are deduced. The criterions of uniqueness of particular solutions of such matrix equations are established.

1. Preliminaries

1.1. Introduction

The matrix linear equations play a fundamental role in many talks in control and dynamical systems theory [1โ€“4]. The such equations are the matrix linear bilateral equations with one and two variables๐ด๐‘‹+๐‘‹๐ต=๐ถ,(1.1)๐ด๐‘‹+๐‘Œ๐ต=๐ถ,(1.2) and the matrix linear unilateral equations๐ด๐‘‹+๐ต๐‘Œ=๐ถ,(1.3) where ๐ด, ๐ต, and ๐ถ are matrices of appropriate size over a certain field โ„ฑ or over a ring โ„›, ๐‘‹,๐‘Œ are unknown matrices. Equations (1.1), (1.2) are called Sylvester equations. The equation ๐ด๐‘‹+๐‘‹๐ด๐‘‡=๐ถ, where matrix ๐ด๐‘‡ is transpose of ๐ด, is called Lyapunov equation and it is special case of Sylvester equation. Equation (1.3) is called the matrix linear Diophantine equation [3, 4].

Roth [5] established the criterions of solvability of matrix equations (1.1), (1.2) whose coefficients ๐ด, ๐ต, and ๐ถ are the matrices over a field โ„ฑ.

Theorem 1.1 ([5]). The matrix equation (1.1), where ๐ด, ๐ต, and ๐ถ are matrices with elements in a field โ„ฑ, has a solution ๐‘‹ with elements in โ„ฑ if and only if the matrices โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐‘€=๐ด๐ถ0๐ต,๐‘=๐ด00๐ต(1.4) are similar.
The matrix equation (1.2), where ๐ด, ๐ต, and ๐ถ are matrices with elements in โ„ฑ, has solution ๐‘‹, ๐‘Œ with elements in โ„ฑ if and only if the matrices ๐‘€and ๐‘ are equivalent.

The Rothโ€™s theorem was extended by many authors to the matrix equations (1.1), (1.2) in cases where their coefficients are the matrices over principal ideal rings [6โ€“8], over arbitrary commutative rings [9], and over other rings [10โ€“14].

The matrix linear unilateral equation (1.3) has a solution if and only if one of the following conditions holds:(a)a greatest common left divisor of the matrices ๐ด and B is a left divisor of the matrix ๐ถ;(b)the matrices โ€–๐ด๐ต๐ถโ€– and โ€–๐ด๐ต0โ€– are right equivalent.

In the case where ๐ด, ๐ต, and ๐ถ in (1.3) are the matrices over a polynomial ring โ„ฑ[๐œ†], where โ„ฑ is a field, these conditions were formulated in [1, 4]. It is not difficult to show, that these conditions of solvability hold for the matrix linear unilateral equation (1.3) over a commutative Bezout domain.

The matrix equations (1.1), (1.2), (1.3), where the coefficients ๐ด, ๐ต, and ๐ถ are the matrices over a field โ„ฑ, reduce by means of the Kronecker product to equivalent systems of linear equations [15]. Hence (1.1) over algebraic closed field has unique solution if and only if the matrices ๐ด and โˆ’๐ต have no common characteristic roots.

One of the methods of solving matrix polynomial equation,๐ด(๐œ†)๐‘‹(๐œ†)+๐‘Œ(๐œ†)๐ต(๐œ†)=๐ถ(๐œ†),(1.5) where ๐ด(๐œ†), ๐ต(๐œ†), and ๐ถ(๐œ†) are matrices over a polynomial ring โ„ฑ[๐œ†], is based on reducibility of polynomial equation to equivalent equation with matrix coefficients over a field โ„ฑ, that is,๐ด๐‘+๐‘๐ต=๐ท,(1.6) where ๐ด and ๐ต are companion matrices of matrix polynomials โˆ‘๐ด(๐œ†)=๐‘Ÿ๐‘–=0๐ด๐‘–๐œ†๐‘Ÿโˆ’๐‘– and โˆ‘๐ต(๐œ†)=๐‘ ๐‘—=0๐ต๐‘—๐œ†๐‘ โˆ’๐‘–, respectively, ๐ท is matrix over a field โ„ฑ, and ๐‘ is unknown matrix [16, 17].

Equation (1.5) has a unique solution ๐‘‹0(๐œ†), ๐‘Œ0(๐œ†) of bounded degree deg๐‘‹0(๐œ†)<deg๐ต(๐œ†) if and only if (det๐ด(๐œ†),det๐ต(๐œ†))=1 [17].

Feinstein and Bar-Ness [18] established that for (1.5), in which at least one from the matrix coefficients ๐ด(๐œ†) or ๐ต(๐œ†) is regular, there exists unique minimal solution ๐‘‹0(๐œ†), ๐‘Œ0(๐œ†), such that deg๐‘‹0(๐œ†)<deg๐ต(๐œ†), deg๐‘Œ0(๐œ†)<deg๐ด(๐œ†), if and only if (det๐ด(๐œ†),det๐ต(๐œ†))=1 and deg๐ถ(๐œ†)โ‰คdeg๐ด(๐œ†)+deg๐ต(๐œ†)โˆ’1. The similar result was established in [19] in the case where at least one from matrix coefficients ๐ด(๐œ†) or ๐ต(๐œ†) is regularizable.

In this paper we propose the method of solving matrix linear equations (1.2), (1.3) over a commutative Bezout domain. This method is based on the use of standard form of a pair of matrices with respect to generalized equivalence introduced in [20, 21], and on congruences. We introduce the notion of particular solutions of such matrix equations. We establish the criterions of uniqueness of particular solutions and write down the formulas of general solutions of such equations.

1.2. The Linear Congruences and Diophantine Equations

Let โ„› be a commutative Bezout domain. A commutative domain โ„› is called a Bezout domain if any two elements ๐‘Ž, ๐‘โˆˆโ„› have a greatest common divisor (๐‘Ž,๐‘)=๐‘‘,โ€‰โ€‰๐‘‘โˆˆโ„› and ๐‘‘=๐‘๐‘Ž+๐‘ž๐‘, for some ๐‘, ๐‘žโˆˆโ„› [22, 23]. Note that a commutative domain โ„› is a Bezout domain if and only if any finitely generated ideal is principal.

Further, ๐‘ˆ(โ„›) denotes a group of units of โ„›, โ„›๐‘š denotes a complete set of residues modulo the ideal (๐‘š) generated by element ๐‘šโˆˆโ„› or a complete set of residues modulo ๐‘š. An element ๐‘Ž of โ„› is said to be an associate to an element ๐‘ of โ„›, if ๐‘Ž=๐‘๐‘ข, where ๐‘ข belongs to ๐‘ˆ(โ„›). A set of elements of โ„›, one from each associate class, is said to be a complete set of nonassociates, which we denoted by โ„›โ€ฒ [24]. For example, if โ„›=โ„ค is a ring of integers, then โ„คโ€ฒ can be chosen as the set of positive integers with zero, that is, โ„คโ€ฒ={0,1,2,โ€ฆ}, and โ„ค๐‘š can be chosen as the set of the smallest nonnegative residues, that is, โ„ค๐‘š={0,1,2,โ€ฆ,๐‘šโˆ’1}.

Many properties of divisibility in principal ideal rings [24โ€“27] can be easily generalized to the commutative Bezout domain. Recall some of them which will be used later.

In what follows, โ„› will always denote a commutative Bezout domain.

Lemma 1.2. Each residue class ๐‘Ž(mod๐‘š) over โ„› can be represented as union ๎š๐‘Ž(mod๐‘š)=๐‘Ÿโˆˆ๐‘…๐‘‘๎€ท๎€ธ๐‘Ž+๐‘š๐‘Ÿ(mod๐‘š๐‘‘),(1.7) where the union is taken over all residues of arbitrary complete set of residues โ„›๐‘‘ modulo ๐‘‘, where ๐‘‘โ‰ 0.

In the case where โ„› is an euclidean ring, this lemma was proved in [27]. By the same way, this lemma can be proved in the case where โ„› is a commutative Bezout domain.

The class of elements ๐‘ฅโ‰ก๐‘ฅ0(mod๐‘š) satisfying the congruence ๐‘Ž๐‘ฅโ‰ก๐‘(mod๐‘š) is called solution of this congruence.

Lemma 1.3. Let ๐‘Ž๐‘ฅโ‰ก๐‘(mod๐‘š)(1.8) and (๐‘Ž,๐‘š)=๐‘‘, where ๐‘Ž,๐‘, ๐‘š, ๐‘‘โˆˆโ„›. Congruence (1.8) has a solution if and only if ๐‘‘โˆฃ๐‘,thatis,๐‘=๐‘1๐‘‘.
Let ๐‘Ž=๐‘Ž1๐‘‘, ๐‘=๐‘1๐‘‘,๐‘š=๐‘š1๐‘‘, where (๐‘Ž1,๐‘š1)=1, and ๐‘ฅโ‰ก๐‘ฅ0(mod๐‘š1) be a solution of congruence ๐‘Ž1๐‘ฅโ‰ก๐‘1๎€ทmod๐‘š1๎€ธ.(1.9) Then the general solution of congruence (1.8) has the form: ๐‘ฅโ‰ก๐‘ฅ0+๐‘š1๐‘Ÿ(mod๐‘š),(1.10) where ๐‘Ÿ is any element of โ„›๐‘‘.

Proof. Necessity. It is obvious.
Sufficiency. Let (๐‘Ž,๐‘š)=๐‘‘ and ๐‘‘โˆฃ๐‘. Then dividing both sides a congruence (1.8) and ๐‘š by ๐‘‘, we get congruence (1.9), where (๐‘Ž1,๐‘š1)=1. There exist elements ๐‘ข, ๐‘ฃ of โ„› such that ๐‘Ž1๐‘ข+๐‘š1๐‘ฃ=1. Thus we have ๐‘Ž1๐‘ขโ‰ก1(mod๐‘š1). Multiply two sides of this congruence by ๐‘1โ‰ 0,thatis, ๐‘Ž1๐‘ข๐‘1โ‰ก๐‘1(mod๐‘š1). Therefore, ๐‘ฅโ‰ก๐‘ข๐‘1(mod๐‘š1) is a solution of congruence (1.9). Set ๐‘ข๐‘1=๐‘ฅ0. Then by Lemma 1.2 we get the general solution of congruence (1.8): ๐‘ฅโ‰ก๐‘ฅ0+๐‘š1๐‘Ÿ(mod๐‘š), where ๐‘Ÿ is an arbitrary element of โ„›๐‘‘. This proves the lemma.

Corollary 1.4. The congruence (1.8) has unique solution ๐‘ฅโ‰ก๐‘ฅ0(mod๐‘š) such that ๐‘ฅ0โˆˆโ„›๐‘š if and only if (๐‘Ž,๐‘š)=1.

Let๐‘Ž๐‘ฅ+๐‘๐‘ฆ=๐‘(1.11) be a linear Diophantine equation over โ„› and (๐‘Ž,๐‘)=๐‘‘. Equation (1.11) has a solution if and only if ๐‘‘โˆฃ๐‘.

Suppose that ๐‘Ž=๐‘Ž1๐‘‘, ๐‘=๐‘1๐‘‘, and ๐‘=๐‘1๐‘‘, where (๐‘Ž1,๐‘1)=1. Then (1.11) implies๐‘Ž1๐‘ฅ+๐‘1๐‘ฆ=๐‘1.(1.12) Let ๐‘ฅ0, ๐‘ฆ0 be a solution of (1.12), that is, ๐‘ฅ0 is a solution of congruence ๐‘Ž1๐‘ฅโ‰ก๐‘1(mod๐‘1), ๐‘ฅ0โˆˆโ„›๐‘1, and ๐‘ฆ0=(๐‘1โˆ’๐‘Ž1๐‘ฅ0)/๐‘1. It is easily to verify that if ๐‘ฅ0โˆˆโ„›๐‘1, then ๐‘ฅ0โˆˆโ„›๐‘.

The solution ๐‘ฅ0, ๐‘ฆ0 of (1.12) is obviously the solution of (1.11).

Definition 1.5. The solution ๐‘ฅ0, ๐‘ฆ0 of (1.11) such that ๐‘ฅ0โˆˆโ„›๐‘ is called the particular solution of this equation.

Then by Lemmas 1.2 and 1.3, a general solution of (1.11) has the form๐‘ฅ=๐‘ฅ0+๐‘๐‘‘๐‘Ÿ+๐‘๐‘˜,๐‘ฆ=๐‘ฆ0โˆ’๐‘Ž๐‘‘๐‘Ÿโˆ’๐‘Ž๐‘˜,(1.13) where ๐‘Ÿ is an arbitrary element of โ„›๐‘‘ and ๐‘˜ is any element of โ„›.

Corollary 1.6. A particular solution ๐‘ฅ0, ๐‘ฆ0 of (1.11) with ๐‘ฅ0โˆˆโ„›๐‘ is unique if and only if (๐‘Ž,๐‘)=1.

Example 1.7. Let 9๐‘ฅ+6๐‘ฆ=6(1.14) be linear Diophantine equation over ring โ„ค. Then (9,6)=3=๐‘‘ and (1.14) is solvable. Then the particular solutions of (1.14) are ๐‘ฅ1(0)=0,๐‘ฆ1(0)๐‘ฅ=1,2(0)=2,๐‘ฆ2(0)๐‘ฅ=โˆ’2,3(0)=4,๐‘ฆ3(0)=โˆ’5,(1.15) because ๐‘ฅ1(0), ๐‘ฅ2(0),๐‘ฅ3(0)โˆˆโ„ค6={0,1,2,3,4,5}.
Then the general solution of (1.14) can be written using (1.13) 6๐‘ฅ=0+39๐‘Ÿ+6๐‘˜,๐‘ฆ=1โˆ’3๐‘Ÿโˆ’9๐‘˜,(1.16) that is, ๐‘ฅ=2๐‘Ÿ+6๐‘˜,๐‘ฆ=1โˆ’3๐‘Ÿโˆ’9๐‘˜,(1.17) where ๐‘Ÿ is arbitrary element of โ„ค3={0,1,2} and ๐‘˜ is any element of โ„ค.

1.3. Standard Form of a Pair of Matrices

Let โ„› be a commutative Bezout domain with diagonal reduction of matrices [28], that is, for every matrix ๐ด of the ring of matrices ๐‘€(๐‘›,โ„›), there exist invertible matrices ๐‘ˆ, ๐‘‰โˆˆ๐บ๐ฟ(๐‘›,โ„›) such that๐‘ˆ๐ด๐‘‰๐ด=๐ท๐ด๎€ท๐œ‘=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐œ‘๐‘–โˆฃ๐œ‘๐‘–+1,๐‘–=1,โ€ฆ,๐‘›โˆ’1.(1.18) If ๐œ‘๐‘–โˆˆโ„›โ€ฒ, ๐‘–=1,โ€ฆ,๐‘›, then the matrix ๐ท๐ด is unique and is called the canonical diagonal form (Smith normal form) of the matrix ๐ด. Such rings are so-called adequate rings. The ring โ„› is called an adequate if โ„› is a commutative domain in which every finitely generated ideal is principal and for every ๐‘Ž, ๐‘โˆˆโ„› with ๐‘Žโ‰ 0; ๐‘Ž can be represented as ๐‘Ž=๐‘๐‘‘ where (๐‘,๐‘)=1 and (๐‘‘๐‘–,๐‘)โ‰ 1 for every nonunit factor ๐‘‘๐‘– of ๐‘‘ [29].

Definition 1.8. The pairs (๐ด1,๐ด2) and (๐ต1,๐ต2) of matrices ๐ด๐‘–, ๐ต๐‘–โˆˆ๐‘€(๐‘›,โ„›), ๐‘–=1,2, are called generalized equivalent pairs if ๐ด๐‘–=๐‘ˆ๐ต๐‘–๐‘‰๐‘–, ๐‘–=1,2, for some invertible matrices ๐‘ˆ and ๐‘‰๐‘– over โ„›.

In [20, 21], the forms of the pair of matrices with respect to generalized equivalence are established.

Theorem 1.9. Let โ„› be an adequate ring, and let ๐ด, ๐ตโˆˆ๐‘€(๐‘›,โ„›) be the nonsingular matrices and ๐ท๐ด๎€ท๐œ‘=ฮฆ=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐ท๐ต๎€ท๐œ“=ฮจ=diag1,โ€ฆ,๐œ“๐‘›๎€ธ(1.19) be their canonical diagonal forms.
Then the pair of matrices (๐ด,๐ต) is generalized equivalent to the pair (๐ท๐ด,๐‘‡๐ต), where ๐‘‡๐ต has the following form: ๐‘‡๐ต=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐œ“1๐‘ก0โ‹ฏ021๐œ“1๐œ“2๐‘กโ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘›1๐œ“1๐‘ก๐‘›2๐œ“2โ‹ฏ๐œ“๐‘›โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–,(1.20)๐‘ก๐‘–๐‘—โˆˆโ„›๐›ฟ๐‘–๐‘—, where ๐›ฟ๐‘–๐‘—=(๐œ‘๐‘–/๐œ‘๐‘—,๐œ“๐‘–/๐œ“๐‘—), ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, ๐‘–>๐‘—.

The pair (๐ท๐ด,๐‘‡๐ต) defined in Theorem 1.9 is called the standard form of the pair of matrices (๐ด,๐ต) or the standard pair of matrices (๐ด,๐ต).

Definition 1.10. The pair (๐ด,๐ต) is called diagonalizable if it is generalized equivalent to the pair of diagonal matrices (๐ท๐ด,๐ท๐ต), that is, its standard form is the pair of diagonal matrices (๐ท๐ด,๐ท๐ต).

Corollary 1.11. Let ๐ด, ๐ตโˆˆ๐‘€(๐‘›,โ„›). If (๐œ‘๐‘›/๐œ‘1,๐œ“๐‘›/๐œ“1)=1, then the pair of matrices (๐ด,๐ต) is diagonalizable.

It is clear taking into account Corollary 1.11 that if (det๐ด,det๐ต)=1, then the standard form of matrices (๐ด,๐ต) is the pair of diagonal matrices (๐ท๐ด,๐ท๐ต).

Let us formulate the criterion of diagonalizability of the pair of matrices.

Theorem 1.12. Let ๐ด, ๐ตโˆˆ๐‘€(๐‘›,โ„›) and ๐ด be a nonsingular matrix. Then the pair of matrices (๐ด,๐ต) is generalized equivalent to the pair of diagonal matrices (๐ท๐ด,๐ท๐ต) if and only if the matrices (adj๐ด)๐ต and (adj๐ท๐ด)๐ท๐ต are equivalent, where adj๐ด is an adjoint matrix.

2. The Matrix Linear Unilateral Equations ๐ด๐‘‹+๐ต๐‘Œ=๐ถ

2.1. The Construction of the Solutions of the Matrix Linear Unilateral Equations with Two Variables

Suppose that the matrix linear unilateral equation (1.3) is solvable, and let (๐ท๐ด,๐‘‡๐ต) be a standard form of a pair of matrices (๐ด,๐ต) from (1.3) with respect to generalized equivalence, that is,๐ท๐ด=ฮฆ=๐‘ˆ๐ด๐‘‰๐ด๎€ท๐œ‘=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐‘‡๐ต=๐‘ˆ๐ต๐‘‰๐ต=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐œ“1๐‘ก0โ‹ฏ021๐œ“1๐œ“2๐‘กโ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘›1๐œ“1๐‘ก๐‘›2๐œ“2โ‹ฏ๐œ“๐‘›โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–(2.1) is a lower triangular matrix of the form (1.20) with the principal diagonal๐ท๐ต๎€ท๐œ“=ฮจ=diag1,โ€ฆ,๐œ“๐‘›๎€ธ,(2.2) where ๐‘ˆ, ๐‘‰๐ด, ๐‘‰๐ตโˆˆ๐บ๐ฟ(๐‘›,โ„›).

Then (1.3) is equivalent to the equation๐ท๐ด๎‚๐‘‹+๐‘‡๐ต๎‚๎‚๐‘Œ=๐ถ,(2.3) where ๎‚๐‘‹=๐‘‰๐ดโˆ’1๐‘‹, ๎‚๐‘Œ=๐‘‰๐ตโˆ’1๐‘Œ, and ๎‚๐ถ=๐‘ˆ๐ถ.

The pair of matrices ๎‚๐‘‹0, ๎‚๐‘Œ0 satisfying (2.3) is called the solution of this equation. Then๐‘‹0=๐‘‰๐ด๎‚๐‘‹0,๐‘Œ0=๐‘‰๐ต๎‚๐‘Œ0(2.4) is the solution of (1.3).

The matrix equation (2.3) is equivalent to the system of linear equation:๐œ‘1ฬƒ๐‘ฅ11+๐œ“1ฬƒ๐‘ฆ11=ฬƒ๐‘11,๐œ‘1ฬƒ๐‘ฅ12+๐œ“1ฬƒ๐‘ฆ12=ฬƒ๐‘12,โ‹ฎ๐œ‘1ฬƒ๐‘ฅ1๐‘›+๐œ“1ฬƒ๐‘ฆ1๐‘›=ฬƒ๐‘1๐‘›,๐œ‘2ฬƒ๐‘ฅ21+๐œ“1๐‘ก21ฬƒ๐‘ฆ11+๐œ“2ฬƒ๐‘ฆ21=ฬƒ๐‘21,โ‹ฎ๐œ‘๐‘›ฬƒ๐‘ฅ๐‘›๐‘›+๐œ“1๐‘ก๐‘›1ฬƒ๐‘ฆ1๐‘›+โ‹ฏ+๐œ“๐‘›โˆ’1๐‘ก๐‘›,๐‘›โˆ’1ฬƒ๐‘ฆ๐‘›โˆ’1,๐‘›+๐œ“๐‘›ฬƒ๐‘ฆ๐‘›๐‘›=ฬƒ๐‘๐‘›๐‘›,(2.5) with the variables ฬƒ๐‘ฅ๐‘–๐‘—, ฬƒ๐‘ฆ๐‘–๐‘—, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, where ๐‘ก๐‘–๐‘—, ๐‘–,๐‘—=1,โ€ฆ,๐‘›, from (2.3), or๐œ‘๐‘–ฬƒ๐‘ฅ๐‘–๐‘—+๐‘–โˆ’1๎“๐‘™=1๐œ“๐‘™๐‘ก๐‘–๐‘™ฬƒ๐‘ฆ๐‘™๐‘—+๐œ“๐‘–ฬƒ๐‘ฆ๐‘–๐‘—=ฬƒ๐‘๐‘–๐‘—,๐‘–,๐‘—=1,โ€ฆ,๐‘›,(2.6) where ๎‚๐‘‹=โ€–ฬƒ๐‘ฅ๐‘–๐‘—โ€–๐‘›1, ๎‚๐‘Œ=โ€–ฬƒ๐‘ฆ๐‘–๐‘—โ€–๐‘›1, and ๎‚๐ถ=โ€–ฬƒ๐‘๐‘–๐‘—โ€–๐‘›1.

The solving of this system reduces to the successive solving of linear Diophantine equations of the form๐œ‘๐‘–ฬƒ๐‘ฅ๐‘–๐‘—+๐œ“๐‘–ฬƒ๐‘ฆ๐‘–๐‘—=ฬƒ๐‘๐‘–๐‘—.(2.7)

Using solutions of system (2.6), we construct the solutions ๎‚๐‘‹, ๎‚๐‘Œ of matrix equation (2.3). Then ๐‘‹=๐‘‰๐ด๎‚๐‘‹ and ๐‘Œ=๐‘‰๐ต๎‚๐‘Œ are the solutions of matrix equation (1.3).

2.2. The General Solution of the Matrix Equation ๐ด๐‘‹+๐ต๐‘Œ=๐ถ with the Diagonalizable Pair of Matrices (๐ด,๐ต)

Suppose that the pair of matrices (๐ด,๐ต) is diagonalizable, that is, ๐‘ˆ๐ด๐‘‰๐ด=๐ท๐ด๎€ท๐œ‘=ฮฆ=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐‘ˆ๐ต๐‘‰๐ต=๐ท๐ต๎€ท๐œ“=ฮจ=diag1,โ€ฆ,๐œ“๐‘›๎€ธ(2.8) for some matrices ๐‘ˆ, ๐‘‰๐ด, ๐‘‰๐ตโˆˆ๐บ๐ฟ(๐‘›,โ„›).

Then (1.3) is equivalent to the equationฮฆ๎‚๎‚๎‚๐‘‹+ฮจ๐‘Œ=๐ถ,(2.9) where ๎‚๐‘‹=๐‘‰๐ดโˆ’1๐‘‹, ๎‚๐‘Œ=๐‘‰๐ตโˆ’1๐‘Œ, and ๎‚๐ถ=๐‘ˆ๐ถ.

From matrix equation (2.9), we get the system of linear Diophantine equation:๐œ‘๐‘–ฬƒ๐‘ฅ๐‘–๐‘—+๐œ“๐‘–ฬƒ๐‘ฆ๐‘–๐‘—=ฬƒ๐‘๐‘–๐‘—,๐‘–,๐‘—=1,โ€ฆ,๐‘›.(2.10)

Let ฬƒ๐‘ฅ(0)๐‘–๐‘—, ฬƒ๐‘ฆ(0)๐‘–๐‘—, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, be a particular solution of corresponding equation of system (2.10), that is, ฬƒ๐‘ฅ(0)๐‘–๐‘— is the solution of congruence ๐œ‘๐‘–ฬƒ๐‘ฅ๐‘–๐‘—โ‰กฬƒ๐‘๐‘–๐‘—(mod๐œ“๐‘–), ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆ๐‘…๐œ“๐‘–, and ฬƒ๐‘ฆ(0)๐‘–๐‘—=(ฬƒ๐‘๐‘–๐‘—โˆ’๐œ‘๐‘–ฬƒ๐‘ฅ(0)๐‘–๐‘—)/๐œ“๐‘–.

The general solution of corresponding equation of system (2.10) by the formula (1.13) will have the following form:ฬƒ๐‘ฅ๐‘–๐‘—=ฬƒ๐‘ฅ(0)๐‘–๐‘—+๐œ“๐‘–๐‘‘๐‘–๐‘–๐‘Ÿ๐‘–+๐œ“๐‘–๐‘˜๐‘–๐‘—,ฬƒ๐‘ฆ๐‘–๐‘—=ฬƒ๐‘ฆ(0)๐‘–๐‘—โˆ’๐œ‘๐‘–๐‘‘๐‘–๐‘–๐‘Ÿ๐‘–โˆ’๐œ‘๐‘–๐‘˜๐‘–๐‘—,๐‘–,๐‘—=1,โ€ฆ,๐‘›,(2.11) where ๐‘‘๐‘–๐‘–=(๐œ‘๐‘–,๐œ“๐‘–), ๐‘Ÿ๐‘– are arbitrary elements of โ„›๐‘‘๐‘–๐‘–, and ๐‘˜๐‘–๐‘— are any elements of โ„›, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›. The particular solution of matrix equation (2.9) is๎‚๐‘‹0=โ€–โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–โ€–๐‘›1,๎‚๐‘Œ0=โ€–โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–โ€–๐‘›1,(2.12) where ฬƒ๐‘ฅ(0)๐‘–๐‘—, ฬƒ๐‘ฆ(0)๐‘–๐‘—, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, is a particular solution of corresponding equation of system (2.10). Then๐‘‹0=๐‘‰๐ด๎‚๐‘‹0,๐‘Œ0=๐‘‰๐ต๎‚๐‘Œ0(2.13) is a particular solution of matrix equation (1.3).

Thus we get the following theorem.

Theorem 2.1. Let the pair of matrices (๐ด,๐ต) from matrix equation (1.3) be diagonalizable and its standard pair be the pair of matrices (ฮฆ,ฮจ) in the form (2.8). Let ๎‚๐‘‹0, ๎‚๐‘Œ0 be a particular solution of matrix equation (2.9). Then the general solution of matrix equation (2.9) is ๎‚๎‚๐‘‹๐‘‹=0๎‚ต๐œ“+diag1๐‘‘11๐‘Ÿ1๐œ“,โ€ฆ,๐‘›๐‘‘๐‘›๐‘›๐‘Ÿ๐‘›๎‚ถ๎‚๎‚๐‘Œ๐ฟ+ฮจ๐พ,๐‘Œ=0๎‚ต๐œ‘โˆ’diag1๐‘‘11๐‘Ÿ1๐œ‘,โ€ฆ,๐‘›๐‘‘๐‘›๐‘›๐‘Ÿ๐‘›๎‚ถ๐ฟโˆ’ฮฆ๐พ,(2.14) where ๐‘‘๐‘–๐‘–=(๐œ‘๐‘–,๐œ“๐‘–), ๐‘Ÿ๐‘– are arbitrary elements of โ„›๐‘‘๐‘–๐‘–, ๐‘–=1,โ€ฆ,๐‘›; ๐ฟ=โ€–๐‘™๐‘–๐‘—โ€–๐‘›1, ๐‘™๐‘–๐‘—=1, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›; ๐พ=โ€–๐‘˜๐‘–๐‘—โ€–๐‘›1, ๐‘˜๐‘–๐‘— are arbitrary elements of โ„›.
The general solution of matrix equation (1.3) has the form ๐‘‹=๐‘‰๐ด๎‚๐‘‹,๐‘Œ=๐‘‰๐ต๎‚๐‘Œ.(2.15)

Example 2.2. Consider the equation ๐ด๐‘‹+๐ต๐‘Œ=๐ถ,(2.16) for the matrices โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐ด=โˆ’1735โˆ’918,๐ต=26761236,๐ถ=1534615(2.17) are matrices over โ„ค and โ€–โ€–โ€–โ€–โ€–๐‘ฅ๐‘‹=11๐‘ฅ12๐‘ฅ21๐‘ฅ22โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐‘ฆ,๐‘Œ=11๐‘ฆ12๐‘ฆ21๐‘ฆ22โ€–โ€–โ€–โ€–โ€–(2.18) are unknown matrices.
The matrix equation (2.16) is solvable.
The pair of matrices (๐ด,๐ต) from matrix equation (2.16) by Theorem 1.12 is diagonalizable, since the matrices โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–,๎€ท(adj๐ด)๐ต=18โˆ’359โˆ’1726761236481083072adj๐ท๐ด๎€ธ๐ท๐ต=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–900120012180012(2.19) are equivalent. Therefore, ๐‘ˆ๐ด๐‘‰๐ด=๐ท๐ด=ฮฆ=diag(1,9),๐œ‘1=1,๐œ‘2=9,๐‘ˆ๐ต๐‘‰๐ต=๐ท๐ต=ฮจ=diag(2,12),๐œ“1=2,๐œ“2=12,(2.20) where โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐‘ˆ=1โˆ’201,๐‘‰๐ด=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–2111,๐‘‰๐ต=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–3โˆ’2โˆ’11.(2.21)
Then (2.16) is equivalent to the equation ฮฆ๎‚๎‚๎‚๐‘‹+ฮจ๐‘Œ=๐ถ,(2.22) where ๎‚๐‘‹=๐‘‰๐ดโˆ’1โ€–โ€–โ€–โ€–โ€–๐‘‹=ฬƒ๐‘ฅ11ฬƒ๐‘ฅ12ฬƒ๐‘ฅ21ฬƒ๐‘ฅ22โ€–โ€–โ€–โ€–โ€–,๎‚๐‘Œ=๐‘‰๐ตโˆ’1โ€–โ€–โ€–โ€–โ€–๐‘Œ=ฬƒ๐‘ฆ11ฬƒ๐‘ฆ12ฬƒ๐‘ฆ21ฬƒ๐‘ฆ22โ€–โ€–โ€–โ€–โ€–,๎‚โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๐ถ=๐‘ˆ๐ถ=34615.(2.23)
From matrix equation (2.22), we get the system of linear Diophantine equations: ฬƒ๐‘ฅ11+2ฬƒ๐‘ฆ11=3,ฬƒ๐‘ฅ12+2ฬƒ๐‘ฆ12=4,9ฬƒ๐‘ฅ21+12ฬƒ๐‘ฆ21=6,9ฬƒ๐‘ฅ22+12ฬƒ๐‘ฆ22=15.(2.24)
The particular solution of each linear equation of system (2.24) has the following form: ฬƒ๐‘ฅ(0)11=1,ฬƒ๐‘ฆ(0)11=1,ฬƒ๐‘ฅ(0)12=0,ฬƒ๐‘ฆ(0)12=2,ฬƒ๐‘ฅ(0)21=2,ฬƒ๐‘ฆ(0)21=โˆ’1,ฬƒ๐‘ฅ(0)22=3,ฬƒ๐‘ฆ(0)22=โˆ’1.(2.25)
The particular solution of matrix equation (2.22) is ๎‚๐‘‹0=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–,๎‚๐‘Œ10230=โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–12โˆ’1โˆ’1.(2.26)
Then by (2.14) the general solution of matrix equation (2.22) is ๎‚โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–+โ€–โ€–โ€–โ€–โ€–๐‘‹=10232๐‘Ÿ12๐‘Ÿ14๐‘Ÿ24๐‘Ÿ2โ€–โ€–โ€–โ€–โ€–+โ€–โ€–โ€–โ€–โ€–2๐‘˜112๐‘˜1212๐‘˜2112๐‘˜22โ€–โ€–โ€–โ€–โ€–,๎‚โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–โˆ’โ€–โ€–โ€–โ€–โ€–๐‘Ÿ๐‘Œ=12โˆ’1โˆ’11๐‘Ÿ13๐‘Ÿ23๐‘Ÿ2โ€–โ€–โ€–โ€–โ€–โˆ’โ€–โ€–โ€–โ€–โ€–๐‘˜11๐‘˜129๐‘˜219๐‘˜22โ€–โ€–โ€–โ€–โ€–,(2.27) or ๎‚โ€–โ€–โ€–โ€–โ€–๐‘‹=1+2๐‘˜112๐‘˜122+4๐‘Ÿ2+12๐‘˜213+4๐‘Ÿ2+12๐‘˜22โ€–โ€–โ€–โ€–โ€–,๎‚โ€–โ€–โ€–โ€–โ€–๐‘Œ=1โˆ’๐‘˜112โˆ’๐‘˜12โˆ’1โˆ’3๐‘Ÿ2โˆ’9๐‘˜21โˆ’1โˆ’3๐‘Ÿ2โˆ’9๐‘˜22โ€–โ€–โ€–โ€–โ€–,(2.28) where ๐‘Ÿ1 is from โ„ค1={0},๐‘Ÿ2 is arbitrary element of โ„ค3={0,1,2}, and ๐‘˜๐‘–๐‘—, ๐‘–, ๐‘—=1,2, is arbitrary elements of โ„ค.
Finally, the general solution of matrix equation (2.16) is ๐‘‹=๐‘‰๐ด๎‚โ€–โ€–โ€–โ€–โ€–๐‘‹=4+4๐‘Ÿ2+4๐‘˜11+12๐‘˜213+4๐‘Ÿ2+4๐‘˜12+12๐‘˜223+4๐‘Ÿ2+2๐‘˜11+12๐‘˜213+4๐‘Ÿ2+2๐‘˜12+12๐‘˜22โ€–โ€–โ€–โ€–โ€–,๐‘Œ=๐‘‰๐ต๎‚โ€–โ€–โ€–โ€–โ€–๐‘Œ=5+6๐‘Ÿ2โˆ’3๐‘˜11+18๐‘˜218+6๐‘Ÿ2โˆ’3๐‘˜12+18๐‘˜22โˆ’2โˆ’3๐‘Ÿ2+๐‘˜11โˆ’9๐‘˜21โˆ’3โˆ’3๐‘Ÿ2+๐‘˜12โˆ’9๐‘˜22โ€–โ€–โ€–โ€–โ€–.(2.29)

2.3. The Uniqueness of Particular Solutions of the Matrix Linear Unilateral Equation

The conditions of uniqueness of solutions of bounded degree (minimal solutions) of matrix linear polynomial equations (1.5) were found in [16โ€“19]. We present the conditions of uniqueness of particular solutions of matrix linear equation over a commutative Bezout domain โ„›.

Theorem 2.3. The matrix equation (2.3) has a unique particular solution ๎‚๐‘‹0=โ€–โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–โ€–๐‘›1,๎‚๐‘Œ0=โ€–โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–โ€–๐‘›1(2.30) such that ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆโ„›๐œ“๐‘–, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, if and only if (det๐ท๐ด,det๐‘‡๐ต)=1.

Proof. From matrix equation (2.3), we get the system of linear equations (2.6). The solving of this system reduces to the successive solving of the linear Diophantine equations of the form (2.7).
The matrix equation (2.3) has a unique particular solution ๎‚๐‘‹0=โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–๐‘›1, ๎‚๐‘Œ0=โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–๐‘›1 such that ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆโ„›๐œ“๐‘–, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, if and only if each linear Diophantine equations of the form (2.7) has a unique particular solution ฬƒ๐‘ฅ(0)๐‘–๐‘—, ฬƒ๐‘ฆ(0)๐‘–๐‘— such that ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆโ„›๐œ“๐‘–, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›. By Corollary 1.6, this holds if and only if (๐œ‘๐‘–,๐œ“๐‘–)=1 for all ๐‘–=1,โ€ฆ,๐‘›. It follows that (det๐ท๐ด,det๐‘‡๐ต)=1. This completes the proof.

Theorem 2.4. Let ๎‚๐‘‹0=โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–๐‘›1, ๎‚๐‘Œ0=โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–๐‘›1, where ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆโ„›๐œ“๐‘–,๐‘–=1,โ€ฆ,๐‘›, be a unique particular solution of matrix equation (2.3).
Then the general solution of matrix equation (2.3) is ๎‚๎‚๐‘‹๐‘‹=0๎‚๎‚๐‘Œ+ฮจ๐พ,๐‘Œ=0โˆ’ฮฆ๐พ,(2.31) where ฮฆ=๐ท๐ด and ฮจ=๐ท๐ต are canonical diagonal forms of ๐ด and ๐ต from matrix equation (1.3), respectively, ๐พ=โ€–๐‘˜๐‘–๐‘—โ€–๐‘›1, ๐‘˜๐‘–๐‘— are arbitrary elements of โ„›, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›.
The general solution of matrix equation (1.3) is the pair of matrices ๐‘‹=๐‘‰๐ด๎‚๐‘‹,๐‘Œ=๐‘‰๐ต๎‚๐‘Œ.(2.32)

Proof. The particular solution of the form (2.30) of (2.3) is unique if and only if (det๐ท๐ด,det๐‘‡๐ต)=1,that is, (det๐ด,det๐ต)=1. Then by Corollary 1.11 the pair of matrices (๐ด,๐ต) is diagonalizable and (1.3) gives us the equation of the form (2.9).
Thus by Theorem 2.1, we get the formula (2.31) of the general solution of (2.3) and the formula (2.32) for computation of general solution of (1.3) in the case where (2.3) has unique particular solution of the form (2.30). The theorem is proved.

3. The Matrix Linear Bilateral Equations ๐ด๐‘‹+๐‘Œ๐ต=๐ถ

Consider the matrix linear bilateral equation (1.2), where ๐ด, ๐ต, and ๐ถ are matrices over a commutative Bezout domain โ„›, and๐‘ˆ๐ด๐ด๐‘‰๐ด=๐ท๐ด๎€ท๐œ‘=ฮฆ=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐œ‘๐‘–โˆฃ๐œ‘๐‘–+1,๐‘ˆ๐ต๐ต๐‘‰๐ต=๐ท๐ต๎€ท๐œ“=ฮจ=diag1,โ€ฆ,๐œ“๐‘›๎€ธ,๐œ“๐‘–โˆฃ๐œ“๐‘–+1,๐‘–=1,โ€ฆ,๐‘›โˆ’1(3.1) are the canonical diagonal forms of matrices ๐ด and ๐ต, respectively, and ๐‘ˆ๐ด, ๐‘‰๐ด, ๐‘ˆ๐ต, ๐‘‰๐ตโˆˆ๐บ๐ฟ(๐‘›,โ„›).

Then (1.2) is equivalent to ฮฆ๎‚๎‚๎‚๐‘‹+๐‘Œฮจ=๐ถ,(3.2) where ๎‚๐‘‹=๐‘‰๐ดโˆ’1๐‘‹๐‘‰๐ต, ๎‚๐‘Œ=๐‘ˆ๐ด๐‘Œ๐‘ˆ๐ตโˆ’1, and ๎‚๐ถ=๐‘ˆ๐ด๐ถ๐‘‰๐ต.

Such an approach to solving (1.2), where ๐ด, ๐ต and ๐ถ are the matrices over a polynomial ring โ„ฑ[๐œ†], where โ„ฑ is a field, was applied in [3].

The equation (3.2) is equivalent to the system of linear Diophantine equations๐œ‘๐‘–ฬƒ๐‘ฅ๐‘–๐‘—+๐œ“๐‘—ฬƒ๐‘ฆ๐‘–๐‘—=ฬƒ๐‘๐‘–๐‘—,๐‘–,๐‘—=1,โ€ฆ,๐‘›.(3.3)

Theorem 3.1. Let ๎‚๐‘‹0=โ€–โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–โ€–๐‘›1,๎‚๐‘Œ0=โ€–โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–โ€–๐‘›1(3.4) be a particular solution of matrix equation (3.2), that is, ฬƒ๐‘ฅ(0)๐‘–๐‘—, ฬƒ๐‘ฆ(0)๐‘–๐‘—, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›, are particular solutions of linear Diophantine equations of system (3.3).
The general solution of matrix equation (3.2) is ๎‚๎‚๐‘‹๐‘‹=0+๐‘Šฮจ๎‚๎‚๐‘Œ+๐พฮจ,๐‘Œ=0โˆ’๐‘Šฮฆโˆ’๐พฮฆ,(3.5) where ๐‘Šฮจ=โ€–(๐œ“๐‘—/๐‘‘๐‘–๐‘—)๐‘ค๐‘–๐‘—โ€–๐‘›1, ๐‘Šฮฆ=โ€–(๐œ‘๐‘—/๐‘‘๐‘–๐‘—)๐‘ค๐‘–๐‘—โ€–๐‘›1, where ๐‘ค๐‘–๐‘— are arbitrary elements of โ„›๐‘‘๐‘–๐‘—, and ๐พ=โ€–๐‘˜๐‘–๐‘—โ€–๐‘›1, where ๐‘˜๐‘–๐‘— are arbitrary elements of โ„›, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›.
The general solution of matrix equation (1.2) is ๐‘‹=๐‘‰๐ด๎‚๐‘‹๐‘‰๐ตโˆ’1,๐‘Œ=๐‘ˆ๐ดโˆ’1๎‚๐‘Œ๐‘ˆ๐ต.(3.6)

Similarly as for (2.3), we prove that particular solution of (3.2) is unique if and only if (detฮฆ,detฮจ)=1. Then by the same way as for (1.3) we write down the general solution of matrix equation (1.2).

Theorem 3.2. Suppose that ๎‚๐‘‹0=โ€–ฬƒ๐‘ฅ(0)๐‘–๐‘—โ€–๐‘›1 and ๎‚๐‘Œ0=โ€–ฬƒ๐‘ฆ(0)๐‘–๐‘—โ€–๐‘›1, where ฬƒ๐‘ฅ(0)๐‘–๐‘—โˆˆโ„›๐œ“๐‘–, ๐‘–=1,โ€ฆ,๐‘›, is unique particular solution of matrix equation (3.2) and ๐ท๐ด๎€ท๐œ‘=ฮฆ=diag1,โ€ฆ,๐œ‘๐‘›๎€ธ,๐ท๐ต๎€ท๐œ“=ฮจ=diag1,โ€ฆ,๐œ“๐‘›๎€ธ(3.7) are canonical diagonal forms of matrices ๐ด, ๐ต from matrix equation (1.2), respectively.
Then the general solution of matrix equation (3.2) is ๎‚๎‚๐‘‹๐‘‹=0๎‚๎‚๐‘Œ+๐พฮจ,๐‘Œ=0โˆ’๐พฮฆ,(3.8) where ๐พ=โ€–๐‘˜๐‘–๐‘—โ€–๐‘›1; ๐‘˜๐‘–๐‘— are arbitrary elements of โ„›, ๐‘–, ๐‘—=1,โ€ฆ,๐‘›.
The general solution of matrix equation (1.2) is ๐‘‹=๐‘‰๐ด๎‚๐‘‹๐‘‰๐ตโˆ’1,๐‘Œ=๐‘ˆ๐ดโˆ’1๎‚๐‘Œ๐‘ˆ๐ต.(3.9)

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