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

Authors: N. S. Dzhaliuk; V. M. Petrychkovych
Journal: ISRN Algebra
 (2012)

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## Abstract

The method of solving matrix linear equationsAX+BY=C and AX+YB=C 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.


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## Body

## 1. Preliminaries

### 1.1. Introduction

The matrix linear equations play a fundamental role in many talks in control and dynamical systems theory [1–4]. The such equations are the matrix linear bilateral equations with one and two variables(1.1)AX+XB=C,(1.2)AX+YB=C,
and the matrix linear unilateral equations(1.3)AX+BY=C,
where A, B, and C are matrices of appropriate size over a certain field ℱ or over a ring ℛ, X,Y are unknown matrices. Equations (1.1), (1.2) are called Sylvester equations. The equation AX+XAT=C, where matrix AT is transpose of A, 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 A, B, and C are the matrices over a field ℱ.Theorem 1.1 ([5]).
The matrix equation (1.1), where A, B, and C are matrices with elements in a field ℱ, has a solution X with elements in ℱ if and only if the matrices
(1.4)M=‖AC0B‖,N=‖A00B‖
are similar.
The matrix equation (1.2), where A, B, and C are matrices with elements in ℱ, has solution X, Y with elements in ℱ if and only if the matrices Mand N are equivalent.The Roth’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 [6–8], over arbitrary commutative rings [9], and over other rings [10–14].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 matricesA and B is a left divisor of the matrix C;(b)
the matrices∥ABC∥ and ∥AB0∥ are right equivalent.In the case whereA, B, and C in (1.3) are the matrices over a polynomial ring ℱ[λ], where ℱ 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 A, B, and C are the matrices over a field ℱ, 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 A and -B have no common characteristic roots.One of the methods of solving matrix polynomial equation,(1.5)A(λ)X(λ)+Y(λ)B(λ)=C(λ),
where A(λ), B(λ), and C(λ) are matrices over a polynomial ring ℱ[λ], is based on reducibility of polynomial equation to equivalent equation with matrix coefficients over a field ℱ, that is,(1.6)AZ+ZB=D,
where A and B are companion matrices of matrix polynomials A(λ)=∑i=0rAiλr-i and B(λ)=∑j=0sBjλs-i, respectively, D is matrix over a field ℱ, and Z is unknown matrix [16, 17].Equation (1.5) has a unique solution X0(λ), Y0(λ) of bounded degree degX0(λ)<degB(λ) if and only if (detA(λ),detB(λ))=1 [17].Feinstein and Bar-Ness [18] established that for (1.5), in which at least one from the matrix coefficients A(λ) or B(λ) is regular, there exists unique minimal solution X0(λ), Y0(λ), such that degX0(λ)<degB(λ), degY0(λ)<degA(λ), if and only if (detA(λ),detB(λ))=1 and degC(λ)≤degA(λ)+degB(λ)-1. The similar result was established in [19] in the case where at least one from matrix coefficients A(λ) or B(λ) 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

Letℛ be a commutative Bezout domain. A commutative domain ℛ is called a Bezout domain if any two elements a, b∈ℛ have a greatest common divisor (a,b)=d,  d∈ℛ and d=pa+qb, for some p, q∈ℛ [22, 23]. Note that a commutative domain ℛ is a Bezout domain if and only if any finitely generated ideal is principal.Further,U(ℛ) denotes a group of units of ℛ, ℛm denotes a complete set of residues modulo the ideal (m) generated by element m∈ℛ or a complete set of residues modulo m. An element a of ℛ is said to be an associate to an element b of ℛ, if a=bu, where u belongs to U(ℛ). A set of elements of ℛ, one from each associate class, is said to be a complete set of nonassociates, which we denoted by ℛ′ [24]. For example, if ℛ=ℤ is a ring of integers, then ℤ′ can be chosen as the set of positive integers with zero, that is, ℤ′={0,1,2,…}, and ℤm can be chosen as the set of the smallest nonnegative residues, that is, ℤm={0,1,2,…,m-1}.Many properties of divisibility in principal ideal rings [24–27] can be easily generalized to the commutative Bezout domain. Recall some of them which will be used later.In what follows,ℛ will always denote a commutative Bezout domain.Lemma 1.2.
Each residue classa¯(modm) over ℛ can be represented as union
(1.7)a¯(mod⁡m)=⋃r∈Rd(a+mr¯)(mod⁡md),
where the union is taken over all residues of arbitrary complete set of residues ℛd modulo d, where d≠0.In the case whereℛ is an euclidean ring, this lemma was proved in [27]. By the same way, this lemma can be proved in the case where ℛ is a commutative Bezout domain.The class of elementsx≡x0(modm) satisfying the congruence ax≡b(modm) is called solution of this congruence.Lemma 1.3.
Let(1.8)ax≡b(mod⁡m)
and (a,m)=d, where a,b, m, d∈ℛ. Congruence (1.8) has a solution if and only if d∣b,thatis,b=b1d.
Leta=a1d, b=b1d,m=m1d, where (a1,m1)=1, and x≡x0(modm1) be a solution of congruence
(1.9)a1x≡b1(mod⁡m1).
Then the general solution of congruence (1.8) has the form:
(1.10)x≡x0+m1r(mod⁡m),
where r is any element of ℛd.Proof.
Necessity. It is obvious.
Sufficiency. Let(a,m)=d and d∣b. Then dividing both sides a congruence (1.8) and m by d, we get congruence (1.9), where (a1,m1)=1. There exist elements u, v of ℛ such that a1u+m1v=1. Thus we have a1u≡1(modm1). Multiply two sides of this congruence by b1≠0,thatis, a1ub1≡b1(modm1). Therefore, x≡ub1(modm1) is a solution of congruence (1.9). Set ub1=x0. Then by Lemma 1.2 we get the general solution of congruence (1.8): x≡x0+m1r(modm), where r is an arbitrary element of ℛd. This proves the lemma.Corollary 1.4.
The congruence (1.8) has unique solution x≡x0(modm) such that x0∈ℛm if and only if (a,m)=1.Let(1.11)ax+by=c
be a linear Diophantine equation over ℛ and (a,b)=d. Equation (1.11) has a solution if and only if d∣c.Suppose thata=a1d, b=b1d, and c=c1d, where (a1,b1)=1. Then (1.11) implies(1.12)a1x+b1y=c1.
Let x0, y0 be a solution of (1.12), that is, x0 is a solution of congruence a1x≡c1(modb1), x0∈ℛb1, and y0=(c1-a1x0)/b1. It is easily to verify that if x0∈ℛb1, then x0∈ℛb.The solutionx0, y0 of (1.12) is obviously the solution of (1.11).Definition 1.5.
The solutionx0, y0 of (1.11) such that x0∈ℛb is called the particular solution of this equation.Then by Lemmas1.2 and 1.3, a general solution of (1.11) has the form(1.13)x=x0+bdr+bk,y=y0-adr-ak,
where r is an arbitrary element of ℛd and k is any element of ℛ.Corollary 1.6.
A particular solutionx0, y0 of (1.11) with x0∈ℛb is unique if and only if (a,b)=1.Example 1.7.
Let(1.14)9x+6y=6
be linear Diophantine equation over ring ℤ. Then (9,6)=3=d and (1.14) is solvable. Then the particular solutions of (1.14) are
(1.15)x1(0)=0,y1(0)=1,x2(0)=2,y2(0)=-2,x3(0)=4,y3(0)=-5,
because x1(0), x2(0),x3(0)∈ℤ6={0,1,2,3,4,5}.
Then the general solution of (1.14) can be written using (1.13)
(1.16)x=0+63r+6k,y=1-93r-9k,
that is,
(1.17)x=2r+6k,y=1-3r-9k,
where r is arbitrary element of ℤ3={0,1,2} and k is any element of ℤ.


### 1.3. Standard Form of a Pair of Matrices

Letℛ be a commutative Bezout domain with diagonal reduction of matrices [28], that is, for every matrix A of the ring of matrices M(n,ℛ), there exist invertible matrices U, V∈GL(n,ℛ) such that(1.18)UAVA=DA=diag⁡(φ1,…,φn),φi∣φi+1,i=1,…,n-1.
If φi∈ℛ′, i=1,…,n, then the matrix DA is unique and is called the canonical diagonal form (Smith normal form) of the matrix A. Such rings are so-called adequate rings. The ring ℛ is called an adequate if ℛ is a commutative domain in which every finitely generated ideal is principal and for every a, b∈ℛ with a≠0; a can be represented as a=cd where (c,b)=1 and (di,b)≠1 for every nonunit factor di of d [29].Definition 1.8.
The pairs(A1,A2) and (B1,B2) of matrices Ai, Bi∈M(n,ℛ), i=1,2, are called generalized equivalent pairs if Ai=UBiVi, i=1,2, for some invertible matrices U and Vi over ℛ.In [20, 21], the forms of the pair of matrices with respect to generalized equivalence are established.Theorem 1.9.
Letℛ be an adequate ring, and let A, B∈M(n,ℛ) be the nonsingular matrices and
(1.19)DA=Φ=diag⁡(φ1,…,φn),DB=Ψ=diag⁡(ψ1,…,ψn)
be their canonical diagonal forms.
Then the pair of matrices(A,B) is generalized equivalent to the pair (DA,TB), where TB has the following form:
(1.20)TB=‖ψ10⋯0t21ψ1ψ2⋯0⋯⋯⋯⋯tn1ψ1tn2ψ2⋯ψn‖,tij∈ℛδij, where δij=(φi/φj,ψi/ψj), i, j=1,…,n, i>j.The pair(DA,TB) defined in Theorem 1.9 is called the standard form of the pair of matrices (A,B) or the standard pair of matrices (A,B).Definition 1.10.
The pair(A,B) is called diagonalizable if it is generalized equivalent to the pair of diagonal matrices (DA,DB), that is, its standard form is the pair of diagonal matrices (DA,DB).Corollary 1.11.
LetA, B∈M(n,ℛ). If (φn/φ1,ψn/ψ1)=1, then the pair of matrices (A,B) is diagonalizable.It is clear taking into account Corollary1.11 that if (detA,detB)=1, then the standard form of matrices (A,B) is the pair of diagonal matrices (DA,DB).Let us formulate the criterion of diagonalizability of the pair of matrices.Theorem 1.12.
LetA, B∈M(n,ℛ) and A be a nonsingular matrix. Then the pair of matrices (A,B) is generalized equivalent to the pair of diagonal matrices (DA,DB) if and only if the matrices (adjA)B and (adjDA)DB are equivalent, where adjA is an adjoint matrix.


## 1.1. Introduction

The matrix linear equations play a fundamental role in many talks in control and dynamical systems theory [1–4]. The such equations are the matrix linear bilateral equations with one and two variables(1.1)AX+XB=C,(1.2)AX+YB=C,
and the matrix linear unilateral equations(1.3)AX+BY=C,
where A, B, and C are matrices of appropriate size over a certain field ℱ or over a ring ℛ, X,Y are unknown matrices. Equations (1.1), (1.2) are called Sylvester equations. The equation AX+XAT=C, where matrix AT is transpose of A, 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 A, B, and C are the matrices over a field ℱ.Theorem 1.1 ([5]).
The matrix equation (1.1), where A, B, and C are matrices with elements in a field ℱ, has a solution X with elements in ℱ if and only if the matrices
(1.4)M=‖AC0B‖,N=‖A00B‖
are similar.
The matrix equation (1.2), where A, B, and C are matrices with elements in ℱ, has solution X, Y with elements in ℱ if and only if the matrices Mand N are equivalent.The Roth’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 [6–8], over arbitrary commutative rings [9], and over other rings [10–14].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 matricesA and B is a left divisor of the matrix C;(b)
the matrices∥ABC∥ and ∥AB0∥ are right equivalent.In the case whereA, B, and C in (1.3) are the matrices over a polynomial ring ℱ[λ], where ℱ 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 A, B, and C are the matrices over a field ℱ, 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 A and -B have no common characteristic roots.One of the methods of solving matrix polynomial equation,(1.5)A(λ)X(λ)+Y(λ)B(λ)=C(λ),
where A(λ), B(λ), and C(λ) are matrices over a polynomial ring ℱ[λ], is based on reducibility of polynomial equation to equivalent equation with matrix coefficients over a field ℱ, that is,(1.6)AZ+ZB=D,
where A and B are companion matrices of matrix polynomials A(λ)=∑i=0rAiλr-i and B(λ)=∑j=0sBjλs-i, respectively, D is matrix over a field ℱ, and Z is unknown matrix [16, 17].Equation (1.5) has a unique solution X0(λ), Y0(λ) of bounded degree degX0(λ)<degB(λ) if and only if (detA(λ),detB(λ))=1 [17].Feinstein and Bar-Ness [18] established that for (1.5), in which at least one from the matrix coefficients A(λ) or B(λ) is regular, there exists unique minimal solution X0(λ), Y0(λ), such that degX0(λ)<degB(λ), degY0(λ)<degA(λ), if and only if (detA(λ),detB(λ))=1 and degC(λ)≤degA(λ)+degB(λ)-1. The similar result was established in [19] in the case where at least one from matrix coefficients A(λ) or B(λ) 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

Letℛ be a commutative Bezout domain. A commutative domain ℛ is called a Bezout domain if any two elements a, b∈ℛ have a greatest common divisor (a,b)=d,  d∈ℛ and d=pa+qb, for some p, q∈ℛ [22, 23]. Note that a commutative domain ℛ is a Bezout domain if and only if any finitely generated ideal is principal.Further,U(ℛ) denotes a group of units of ℛ, ℛm denotes a complete set of residues modulo the ideal (m) generated by element m∈ℛ or a complete set of residues modulo m. An element a of ℛ is said to be an associate to an element b of ℛ, if a=bu, where u belongs to U(ℛ). A set of elements of ℛ, one from each associate class, is said to be a complete set of nonassociates, which we denoted by ℛ′ [24]. For example, if ℛ=ℤ is a ring of integers, then ℤ′ can be chosen as the set of positive integers with zero, that is, ℤ′={0,1,2,…}, and ℤm can be chosen as the set of the smallest nonnegative residues, that is, ℤm={0,1,2,…,m-1}.Many properties of divisibility in principal ideal rings [24–27] can be easily generalized to the commutative Bezout domain. Recall some of them which will be used later.In what follows,ℛ will always denote a commutative Bezout domain.Lemma 1.2.
Each residue classa¯(modm) over ℛ can be represented as union
(1.7)a¯(mod⁡m)=⋃r∈Rd(a+mr¯)(mod⁡md),
where the union is taken over all residues of arbitrary complete set of residues ℛd modulo d, where d≠0.In the case whereℛ is an euclidean ring, this lemma was proved in [27]. By the same way, this lemma can be proved in the case where ℛ is a commutative Bezout domain.The class of elementsx≡x0(modm) satisfying the congruence ax≡b(modm) is called solution of this congruence.Lemma 1.3.
Let(1.8)ax≡b(mod⁡m)
and (a,m)=d, where a,b, m, d∈ℛ. Congruence (1.8) has a solution if and only if d∣b,thatis,b=b1d.
Leta=a1d, b=b1d,m=m1d, where (a1,m1)=1, and x≡x0(modm1) be a solution of congruence
(1.9)a1x≡b1(mod⁡m1).
Then the general solution of congruence (1.8) has the form:
(1.10)x≡x0+m1r(mod⁡m),
where r is any element of ℛd.Proof.
Necessity. It is obvious.
Sufficiency. Let(a,m)=d and d∣b. Then dividing both sides a congruence (1.8) and m by d, we get congruence (1.9), where (a1,m1)=1. There exist elements u, v of ℛ such that a1u+m1v=1. Thus we have a1u≡1(modm1). Multiply two sides of this congruence by b1≠0,thatis, a1ub1≡b1(modm1). Therefore, x≡ub1(modm1) is a solution of congruence (1.9). Set ub1=x0. Then by Lemma 1.2 we get the general solution of congruence (1.8): x≡x0+m1r(modm), where r is an arbitrary element of ℛd. This proves the lemma.Corollary 1.4.
The congruence (1.8) has unique solution x≡x0(modm) such that x0∈ℛm if and only if (a,m)=1.Let(1.11)ax+by=c
be a linear Diophantine equation over ℛ and (a,b)=d. Equation (1.11) has a solution if and only if d∣c.Suppose thata=a1d, b=b1d, and c=c1d, where (a1,b1)=1. Then (1.11) implies(1.12)a1x+b1y=c1.
Let x0, y0 be a solution of (1.12), that is, x0 is a solution of congruence a1x≡c1(modb1), x0∈ℛb1, and y0=(c1-a1x0)/b1. It is easily to verify that if x0∈ℛb1, then x0∈ℛb.The solutionx0, y0 of (1.12) is obviously the solution of (1.11).Definition 1.5.
The solutionx0, y0 of (1.11) such that x0∈ℛb is called the particular solution of this equation.Then by Lemmas1.2 and 1.3, a general solution of (1.11) has the form(1.13)x=x0+bdr+bk,y=y0-adr-ak,
where r is an arbitrary element of ℛd and k is any element of ℛ.Corollary 1.6.
A particular solutionx0, y0 of (1.11) with x0∈ℛb is unique if and only if (a,b)=1.Example 1.7.
Let(1.14)9x+6y=6
be linear Diophantine equation over ring ℤ. Then (9,6)=3=d and (1.14) is solvable. Then the particular solutions of (1.14) are
(1.15)x1(0)=0,y1(0)=1,x2(0)=2,y2(0)=-2,x3(0)=4,y3(0)=-5,
because x1(0), x2(0),x3(0)∈ℤ6={0,1,2,3,4,5}.
Then the general solution of (1.14) can be written using (1.13)
(1.16)x=0+63r+6k,y=1-93r-9k,
that is,
(1.17)x=2r+6k,y=1-3r-9k,
where r is arbitrary element of ℤ3={0,1,2} and k is any element of ℤ.


## 1.3. Standard Form of a Pair of Matrices

Letℛ be a commutative Bezout domain with diagonal reduction of matrices [28], that is, for every matrix A of the ring of matrices M(n,ℛ), there exist invertible matrices U, V∈GL(n,ℛ) such that(1.18)UAVA=DA=diag⁡(φ1,…,φn),φi∣φi+1,i=1,…,n-1.
If φi∈ℛ′, i=1,…,n, then the matrix DA is unique and is called the canonical diagonal form (Smith normal form) of the matrix A. Such rings are so-called adequate rings. The ring ℛ is called an adequate if ℛ is a commutative domain in which every finitely generated ideal is principal and for every a, b∈ℛ with a≠0; a can be represented as a=cd where (c,b)=1 and (di,b)≠1 for every nonunit factor di of d [29].Definition 1.8.
The pairs(A1,A2) and (B1,B2) of matrices Ai, Bi∈M(n,ℛ), i=1,2, are called generalized equivalent pairs if Ai=UBiVi, i=1,2, for some invertible matrices U and Vi over ℛ.In [20, 21], the forms of the pair of matrices with respect to generalized equivalence are established.Theorem 1.9.
Letℛ be an adequate ring, and let A, B∈M(n,ℛ) be the nonsingular matrices and
(1.19)DA=Φ=diag⁡(φ1,…,φn),DB=Ψ=diag⁡(ψ1,…,ψn)
be their canonical diagonal forms.
Then the pair of matrices(A,B) is generalized equivalent to the pair (DA,TB), where TB has the following form:
(1.20)TB=‖ψ10⋯0t21ψ1ψ2⋯0⋯⋯⋯⋯tn1ψ1tn2ψ2⋯ψn‖,tij∈ℛδij, where δij=(φi/φj,ψi/ψj), i, j=1,…,n, i>j.The pair(DA,TB) defined in Theorem 1.9 is called the standard form of the pair of matrices (A,B) or the standard pair of matrices (A,B).Definition 1.10.
The pair(A,B) is called diagonalizable if it is generalized equivalent to the pair of diagonal matrices (DA,DB), that is, its standard form is the pair of diagonal matrices (DA,DB).Corollary 1.11.
LetA, B∈M(n,ℛ). If (φn/φ1,ψn/ψ1)=1, then the pair of matrices (A,B) is diagonalizable.It is clear taking into account Corollary1.11 that if (detA,detB)=1, then the standard form of matrices (A,B) is the pair of diagonal matrices (DA,DB).Let us formulate the criterion of diagonalizability of the pair of matrices.Theorem 1.12.
LetA, B∈M(n,ℛ) and A be a nonsingular matrix. Then the pair of matrices (A,B) is generalized equivalent to the pair of diagonal matrices (DA,DB) if and only if the matrices (adjA)B and (adjDA)DB are equivalent, where adjA is an adjoint matrix.


## 2. The Matrix Linear Unilateral EquationsAX+BY=C

### 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 (DA,TB) be a standard form of a pair of matrices (A,B) from (1.3) with respect to generalized equivalence, that is,(2.1)DA=Φ=UAVA=diag⁡(φ1,…,φn),TB=UBVB=‖ψ10⋯0t21ψ1ψ2⋯0⋯⋯⋯⋯tn1ψ1tn2ψ2⋯ψn‖
is a lower triangular matrix of the form (1.20) with the principal diagonal(2.2)DB=Ψ=diag⁡(ψ1,…,ψn),
where U, VA, VB∈GL(n,ℛ).Then (1.3) is equivalent to the equation(2.3)DAX̃+TBỸ=C̃,
where X̃=VA-1X, Ỹ=VB-1Y, and C̃=UC.The pair of matricesX̃0, Ỹ0 satisfying (2.3) is called the solution of this equation. Then(2.4)X0=VAX̃0,Y0=VBỸ0
is the solution of (1.3).The matrix equation (2.3) is equivalent to the system of linear equation:(2.5)φ1x̃11+ψ1ỹ11=c̃11,φ1x̃12+ψ1ỹ12=c̃12,⋮φ1x̃1n+ψ1ỹ1n=c̃1n,φ2x̃21+ψ1t21ỹ11+ψ2ỹ21=c̃21,⋮φnx̃nn+ψ1tn1ỹ1n+⋯+ψn-1tn,n-1ỹn-1,n+ψnỹnn=c̃nn,
with the variables x̃ij, ỹij, i, j=1,…,n, where tij, i,j=1,…,n, from (2.3), or(2.6)φix̃ij+∑l=1i-1ψltilỹlj+ψiỹij=c̃ij,i,j=1,…,n,
where X̃=∥x̃ij∥1n, Ỹ=∥ỹij∥1n, and C̃=∥c̃ij∥1n.The solving of this system reduces to the successive solving of linear Diophantine equations of the form(2.7)φix̃ij+ψiỹij=c̃ij.Using solutions of system (2.6), we construct the solutions X̃, Ỹ of matrix equation (2.3). Then X=VAX̃ and Y=VBỸ are the solutions of matrix equation (1.3).


### 2.2. The General Solution of the Matrix EquationAX+BY=C with the Diagonalizable Pair of Matrices (A,B)

Suppose that the pair of matrices(A,B) is diagonalizable, that is,
(2.8)UAVA=DA=Φ=diag⁡(φ1,…,φn),UBVB=DB=Ψ=diag⁡(ψ1,…,ψn)
for some matrices U, VA, VB∈GL(n,ℛ).Then (1.3) is equivalent to the equation(2.9)ΦX̃+ΨỸ=C̃,
where X̃=VA-1X, Ỹ=VB-1Y, and C̃=UC.From matrix equation (2.9), we get the system of linear Diophantine equation:(2.10)φix̃ij+ψiỹij=c̃ij,i,j=1,…,n.Letx̃ij(0), ỹij(0), i, j=1,…,n, be a particular solution of corresponding equation of system (2.10), that is, x̃ij(0) is the solution of congruence φix̃ij≡c̃ij(modψi), x̃ij(0)∈Rψi, and ỹij(0)=(c̃ij-φix̃ij(0))/ψi.The general solution of corresponding equation of system (2.10) by the formula (1.13) will have the following form:(2.11)x̃ij=x̃ij(0)+ψidiiri+ψikij,ỹij=ỹij(0)-φidiiri-φikij,i,j=1,…,n,
where dii=(φi,ψi), ri are arbitrary elements of ℛdii, and kij are any elements of ℛ, i, j=1,…,n. The particular solution of matrix equation (2.9) is(2.12)X̃0=‖x̃ij(0)‖1n,Ỹ0=‖ỹij(0)‖1n,
where x̃ij(0), ỹij(0), i, j=1,…,n, is a particular solution of corresponding equation of system (2.10). Then(2.13)X0=VAX̃0,Y0=VBỸ0
is a particular solution of matrix equation (1.3).Thus we get the following theorem.Theorem 2.1.
Let the pair of matrices(A,B) from matrix equation (1.3) be diagonalizable and its standard pair be the pair of matrices (Φ,Ψ) in the form (2.8). Let X̃0, Ỹ0 be a particular solution of matrix equation (2.9). Then the general solution of matrix equation (2.9) is
(2.14)X̃=X̃0+diag⁡(ψ1d11r1,…,ψndnnrn)L+ΨK,Ỹ=Ỹ0-diag⁡(φ1d11r1,…,φndnnrn)L-ΦK,
where dii=(φi,ψi), ri are arbitrary elements of ℛdii, i=1,…,n; L=∥lij∥1n, lij=1, i, j=1,…,n; K=∥kij∥1n, kij are arbitrary elements of ℛ.
The general solution of matrix equation (1.3) has the form
(2.15)X=VAX̃,Y=VBỸ.Example 2.2.
Consider the equation(2.16)AX+BY=C,
for the matrices
(2.17)A=‖-1735-918‖,B=‖26761236‖,C=‖1534615‖
are matrices over ℤ and
(2.18)X=‖x11x12x21x22‖,Y=‖y11y12y21y22‖
are unknown matrices.
The matrix equation (2.16) is solvable.
The pair of matrices(A,B) from matrix equation (2.16) by Theorem 1.12 is diagonalizable, since the matrices
(2.19)(adjA)B=‖18-359-17‖‖26761236‖=‖481083072‖,(adjDA)DB=‖9001‖‖20012‖=‖180012‖
are equivalent. Therefore,
(2.20)UAVA=DA=Φ=diag⁡(1,9),φ1=1,φ2=9,UBVB=DB=Ψ=diag⁡(2,12),ψ1=2,ψ2=12,
where
(2.21)U=‖1-201‖,VA=‖2111‖,VB=‖3-2-11‖.
Then (2.16) is equivalent to the equation
(2.22)ΦX̃+ΨỸ=C̃,
where
(2.23)X̃=VA-1X=‖x̃11x̃12x̃21x̃22‖,Ỹ=VB-1Y=‖ỹ11ỹ12ỹ21ỹ22‖,C̃=UC=‖34615‖.
From matrix equation (2.22), we get the system of linear Diophantine equations:
(2.24)x̃11+2ỹ11=3,x̃12+2ỹ12=4,9x̃21+12ỹ21=6,9x̃22+12ỹ22=15.
The particular solution of each linear equation of system (2.24) has the following form:
(2.25)x̃11(0)=1,ỹ11(0)=1,x̃12(0)=0,ỹ12(0)=2,x̃21(0)=2,ỹ21(0)=-1,x̃22(0)=3,ỹ22(0)=-1.
The particular solution of matrix equation (2.22) is
(2.26)X̃0=‖1023‖,Ỹ0=‖12-1-1‖.
Then by (2.14) the general solution of matrix equation (2.22) is
(2.27)X̃=‖1023‖+‖2r12r14r24r2‖+‖2k112k1212k2112k22‖,Ỹ=‖12-1-1‖-‖r1r13r23r2‖-‖k11k129k219k22‖,
or
(2.28)X̃=‖1+2k112k122+4r2+12k213+4r2+12k22‖,Ỹ=‖1-k112-k12-1-3r2-9k21-1-3r2-9k22‖,
where r1 is from ℤ1={0},r2 is arbitrary element of ℤ3={0,1,2}, and kij, i, j=1,2, is arbitrary elements of ℤ.
Finally, the general solution of matrix equation (2.16) is
(2.29)X=VAX̃=‖4+4r2+4k11+12k213+4r2+4k12+12k223+4r2+2k11+12k213+4r2+2k12+12k22‖,Y=VBỸ=‖5+6r2-3k11+18k218+6r2-3k12+18k22-2-3r2+k11-9k21-3-3r2+k12-9k22‖.


### 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 [16–19]. We present the conditions of uniqueness of particular solutions of matrix linear equation over a commutative Bezout domain ℛ.Theorem 2.3.
The matrix equation (2.3) has a unique particular solution
(2.30)X̃0=‖x̃ij(0)‖1n,Ỹ0=‖ỹij(0)‖1n
such that x̃ij(0)∈ℛψi, i, j=1,…,n, if and only if (detDA,detTB)=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 X̃0=∥x̃ij(0)∥1n, Ỹ0=∥ỹij(0)∥1n such that x̃ij(0)∈ℛψi, i, j=1,…,n, if and only if each linear Diophantine equations of the form (2.7) has a unique particular solution x̃ij(0), ỹij(0) such that x̃ij(0)∈ℛψi, i, j=1,…,n. By Corollary 1.6, this holds if and only if (φi,ψi)=1 for all i=1,…,n. It follows that (detDA,detTB)=1. This completes the proof.Theorem 2.4.
LetX̃0=∥x̃ij(0)∥1n, Ỹ0=∥ỹij(0)∥1n, where x̃ij(0)∈ℛψi,i=1,…,n, be a unique particular solution of matrix equation (2.3).
Then the general solution of matrix equation (2.3) is
(2.31)X̃=X̃0+ΨK,Ỹ=Ỹ0-ΦK,
where Φ=DA and Ψ=DB are canonical diagonal forms of A and B from matrix equation (1.3), respectively, K=∥kij∥1n, kij are arbitrary elements of ℛ, i, j=1,…,n.
The general solution of matrix equation (1.3) is the pair of matrices
(2.32)X=VAX̃,Y=VBỸ.Proof.
The particular solution of the form (2.30) of (2.3) is unique if and only if (detDA,detTB)=1,that is, (detA,detB)=1. Then by Corollary 1.11 the pair of matrices (A,B) is diagonalizable and (1.3) gives us the equation of the form (2.9).
Thus by Theorem2.1, 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.


## 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 (DA,TB) be a standard form of a pair of matrices (A,B) from (1.3) with respect to generalized equivalence, that is,(2.1)DA=Φ=UAVA=diag⁡(φ1,…,φn),TB=UBVB=‖ψ10⋯0t21ψ1ψ2⋯0⋯⋯⋯⋯tn1ψ1tn2ψ2⋯ψn‖
is a lower triangular matrix of the form (1.20) with the principal diagonal(2.2)DB=Ψ=diag⁡(ψ1,…,ψn),
where U, VA, VB∈GL(n,ℛ).Then (1.3) is equivalent to the equation(2.3)DAX̃+TBỸ=C̃,
where X̃=VA-1X, Ỹ=VB-1Y, and C̃=UC.The pair of matricesX̃0, Ỹ0 satisfying (2.3) is called the solution of this equation. Then(2.4)X0=VAX̃0,Y0=VBỸ0
is the solution of (1.3).The matrix equation (2.3) is equivalent to the system of linear equation:(2.5)φ1x̃11+ψ1ỹ11=c̃11,φ1x̃12+ψ1ỹ12=c̃12,⋮φ1x̃1n+ψ1ỹ1n=c̃1n,φ2x̃21+ψ1t21ỹ11+ψ2ỹ21=c̃21,⋮φnx̃nn+ψ1tn1ỹ1n+⋯+ψn-1tn,n-1ỹn-1,n+ψnỹnn=c̃nn,
with the variables x̃ij, ỹij, i, j=1,…,n, where tij, i,j=1,…,n, from (2.3), or(2.6)φix̃ij+∑l=1i-1ψltilỹlj+ψiỹij=c̃ij,i,j=1,…,n,
where X̃=∥x̃ij∥1n, Ỹ=∥ỹij∥1n, and C̃=∥c̃ij∥1n.The solving of this system reduces to the successive solving of linear Diophantine equations of the form(2.7)φix̃ij+ψiỹij=c̃ij.Using solutions of system (2.6), we construct the solutions X̃, Ỹ of matrix equation (2.3). Then X=VAX̃ and Y=VBỸ are the solutions of matrix equation (1.3).


## 2.2. The General Solution of the Matrix EquationAX+BY=C with the Diagonalizable Pair of Matrices (A,B)

Suppose that the pair of matrices(A,B) is diagonalizable, that is,
(2.8)UAVA=DA=Φ=diag⁡(φ1,…,φn),UBVB=DB=Ψ=diag⁡(ψ1,…,ψn)
for some matrices U, VA, VB∈GL(n,ℛ).Then (1.3) is equivalent to the equation(2.9)ΦX̃+ΨỸ=C̃,
where X̃=VA-1X, Ỹ=VB-1Y, and C̃=UC.From matrix equation (2.9), we get the system of linear Diophantine equation:(2.10)φix̃ij+ψiỹij=c̃ij,i,j=1,…,n.Letx̃ij(0), ỹij(0), i, j=1,…,n, be a particular solution of corresponding equation of system (2.10), that is, x̃ij(0) is the solution of congruence φix̃ij≡c̃ij(modψi), x̃ij(0)∈Rψi, and ỹij(0)=(c̃ij-φix̃ij(0))/ψi.The general solution of corresponding equation of system (2.10) by the formula (1.13) will have the following form:(2.11)x̃ij=x̃ij(0)+ψidiiri+ψikij,ỹij=ỹij(0)-φidiiri-φikij,i,j=1,…,n,
where dii=(φi,ψi), ri are arbitrary elements of ℛdii, and kij are any elements of ℛ, i, j=1,…,n. The particular solution of matrix equation (2.9) is(2.12)X̃0=‖x̃ij(0)‖1n,Ỹ0=‖ỹij(0)‖1n,
where x̃ij(0), ỹij(0), i, j=1,…,n, is a particular solution of corresponding equation of system (2.10). Then(2.13)X0=VAX̃0,Y0=VBỸ0
is a particular solution of matrix equation (1.3).Thus we get the following theorem.Theorem 2.1.
Let the pair of matrices(A,B) from matrix equation (1.3) be diagonalizable and its standard pair be the pair of matrices (Φ,Ψ) in the form (2.8). Let X̃0, Ỹ0 be a particular solution of matrix equation (2.9). Then the general solution of matrix equation (2.9) is
(2.14)X̃=X̃0+diag⁡(ψ1d11r1,…,ψndnnrn)L+ΨK,Ỹ=Ỹ0-diag⁡(φ1d11r1,…,φndnnrn)L-ΦK,
where dii=(φi,ψi), ri are arbitrary elements of ℛdii, i=1,…,n; L=∥lij∥1n, lij=1, i, j=1,…,n; K=∥kij∥1n, kij are arbitrary elements of ℛ.
The general solution of matrix equation (1.3) has the form
(2.15)X=VAX̃,Y=VBỸ.Example 2.2.
Consider the equation(2.16)AX+BY=C,
for the matrices
(2.17)A=‖-1735-918‖,B=‖26761236‖,C=‖1534615‖
are matrices over ℤ and
(2.18)X=‖x11x12x21x22‖,Y=‖y11y12y21y22‖
are unknown matrices.
The matrix equation (2.16) is solvable.
The pair of matrices(A,B) from matrix equation (2.16) by Theorem 1.12 is diagonalizable, since the matrices
(2.19)(adjA)B=‖18-359-17‖‖26761236‖=‖481083072‖,(adjDA)DB=‖9001‖‖20012‖=‖180012‖
are equivalent. Therefore,
(2.20)UAVA=DA=Φ=diag⁡(1,9),φ1=1,φ2=9,UBVB=DB=Ψ=diag⁡(2,12),ψ1=2,ψ2=12,
where
(2.21)U=‖1-201‖,VA=‖2111‖,VB=‖3-2-11‖.
Then (2.16) is equivalent to the equation
(2.22)ΦX̃+ΨỸ=C̃,
where
(2.23)X̃=VA-1X=‖x̃11x̃12x̃21x̃22‖,Ỹ=VB-1Y=‖ỹ11ỹ12ỹ21ỹ22‖,C̃=UC=‖34615‖.
From matrix equation (2.22), we get the system of linear Diophantine equations:
(2.24)x̃11+2ỹ11=3,x̃12+2ỹ12=4,9x̃21+12ỹ21=6,9x̃22+12ỹ22=15.
The particular solution of each linear equation of system (2.24) has the following form:
(2.25)x̃11(0)=1,ỹ11(0)=1,x̃12(0)=0,ỹ12(0)=2,x̃21(0)=2,ỹ21(0)=-1,x̃22(0)=3,ỹ22(0)=-1.
The particular solution of matrix equation (2.22) is
(2.26)X̃0=‖1023‖,Ỹ0=‖12-1-1‖.
Then by (2.14) the general solution of matrix equation (2.22) is
(2.27)X̃=‖1023‖+‖2r12r14r24r2‖+‖2k112k1212k2112k22‖,Ỹ=‖12-1-1‖-‖r1r13r23r2‖-‖k11k129k219k22‖,
or
(2.28)X̃=‖1+2k112k122+4r2+12k213+4r2+12k22‖,Ỹ=‖1-k112-k12-1-3r2-9k21-1-3r2-9k22‖,
where r1 is from ℤ1={0},r2 is arbitrary element of ℤ3={0,1,2}, and kij, i, j=1,2, is arbitrary elements of ℤ.
Finally, the general solution of matrix equation (2.16) is
(2.29)X=VAX̃=‖4+4r2+4k11+12k213+4r2+4k12+12k223+4r2+2k11+12k213+4r2+2k12+12k22‖,Y=VBỸ=‖5+6r2-3k11+18k218+6r2-3k12+18k22-2-3r2+k11-9k21-3-3r2+k12-9k22‖.


## 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 [16–19]. We present the conditions of uniqueness of particular solutions of matrix linear equation over a commutative Bezout domain ℛ.Theorem 2.3.
The matrix equation (2.3) has a unique particular solution
(2.30)X̃0=‖x̃ij(0)‖1n,Ỹ0=‖ỹij(0)‖1n
such that x̃ij(0)∈ℛψi, i, j=1,…,n, if and only if (detDA,detTB)=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 X̃0=∥x̃ij(0)∥1n, Ỹ0=∥ỹij(0)∥1n such that x̃ij(0)∈ℛψi, i, j=1,…,n, if and only if each linear Diophantine equations of the form (2.7) has a unique particular solution x̃ij(0), ỹij(0) such that x̃ij(0)∈ℛψi, i, j=1,…,n. By Corollary 1.6, this holds if and only if (φi,ψi)=1 for all i=1,…,n. It follows that (detDA,detTB)=1. This completes the proof.Theorem 2.4.
LetX̃0=∥x̃ij(0)∥1n, Ỹ0=∥ỹij(0)∥1n, where x̃ij(0)∈ℛψi,i=1,…,n, be a unique particular solution of matrix equation (2.3).
Then the general solution of matrix equation (2.3) is
(2.31)X̃=X̃0+ΨK,Ỹ=Ỹ0-ΦK,
where Φ=DA and Ψ=DB are canonical diagonal forms of A and B from matrix equation (1.3), respectively, K=∥kij∥1n, kij are arbitrary elements of ℛ, i, j=1,…,n.
The general solution of matrix equation (1.3) is the pair of matrices
(2.32)X=VAX̃,Y=VBỸ.Proof.
The particular solution of the form (2.30) of (2.3) is unique if and only if (detDA,detTB)=1,that is, (detA,detB)=1. Then by Corollary 1.11 the pair of matrices (A,B) is diagonalizable and (1.3) gives us the equation of the form (2.9).
Thus by Theorem2.1, 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 EquationsAX+YB=C

Consider the matrix linear bilateral equation (1.2), where A, B, and C are matrices over a commutative Bezout domain ℛ, and(3.1)UAAVA=DA=Φ=diag⁡(φ1,…,φn),φi∣φi+1,UBBVB=DB=Ψ=diag⁡(ψ1,…,ψn),ψi∣ψi+1,i=1,…,n-1
are the canonical diagonal forms of matrices A and B, respectively, and UA, VA, UB, VB∈GL(n,ℛ).Then (1.2) is equivalent to (3.2)ΦX̃+ỸΨ=C̃,
where X̃=VA-1XVB, Ỹ=UAYUB-1, and C̃=UACVB.Such an approach to solving (1.2), where A, B and C are the matrices over a polynomial ring ℱ[λ], where ℱ is a field, was applied in [3].The equation (3.2) is equivalent to the system of linear Diophantine equations(3.3)φix̃ij+ψjỹij=c̃ij,i,j=1,…,n.Theorem 3.1.
Let(3.4)X̃0=‖x̃ij(0)‖1n,Ỹ0=‖ỹij(0)‖1n
be a particular solution of matrix equation (3.2), that is, x̃ij(0), ỹij(0), i, j=1,…,n, are particular solutions of linear Diophantine equations of system (3.3).
The general solution of matrix equation (3.2) is
(3.5)X̃=X̃0+WΨ+KΨ,Ỹ=Ỹ0-WΦ-KΦ,
where WΨ=∥(ψj/dij)wij∥1n, WΦ=∥(φj/dij)wij∥1n, where wij are arbitrary elements of ℛdij, and K=∥kij∥1n, where kij are arbitrary elements of ℛ, i, j=1,…,n.
The general solution of matrix equation (1.2) is
(3.6)X=VAX̃VB-1,Y=UA-1ỸUB.Similarly as for (2.3), we prove that particular solution of (3.2) is unique if and only if (detΦ,detΨ)=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 thatX̃0=∥x̃ij(0)∥1n and Ỹ0=∥ỹij(0)∥1n, where x̃ij(0)∈ℛψi, i=1,…,n, is unique particular solution of matrix equation (3.2) and
(3.7)DA=Φ=diag⁡(φ1,…,φn),DB=Ψ=diag⁡(ψ1,…,ψn)
are canonical diagonal forms of matrices A, B from matrix equation (1.2), respectively.
Then the general solution of matrix equation (3.2) is
(3.8)X̃=X̃0+KΨ,Ỹ=Ỹ0-KΦ,
where K=∥kij∥1n; kij are arbitrary elements of ℛ, i, j=1,…,n.
The general solution of matrix equation (1.2) is
(3.9)X=VAX̃VB-1,Y=UA-1ỸUB.



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*Source: 205478-2012-05-08.xml*