International Journal of Analysis
Volume 2017 (2017), Article ID 2867354, 7 pages
https://doi.org/10.1155/2017/2867354

Research Article

The Structure of Symmetric Solutions of the Matrix Equation over a Principal Ideal Domain

V. M. Prokip

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Naukova Str. 3b, L’viv, Ukraine

Correspondence should be addressed to V. M. Prokip; v.prokip@gmail.com

Received 20 June 2017; Accepted 18 September 2017; Published 5 November 2017

Academic Editor: Fatemeh P. Ali Beik

Copyright © 2017 V. M. Prokip. 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.

We investigate the structure of symmetric solutions of the matrix equation , where and are -by- matrices over a principal ideal domain and is unknown -by- matrix over . We prove that matrix equation over has a symmetric solution if and only if equation has a solution over and the matrix is symmetric. If symmetric solution exists we propose the method for its construction.

1. Introduction

Let be a principal ideal domain with an identity element and let denote the set of matrices over . In what follows is the zero matrix. The transpose matrix of matrix will be denoted by . Matrix is called a symmetric matrix if .

Consider the matrix equationwhere and is unknown matrix over . The problem of solvability of (1) has drawn the attention of many mathematicians. Numerous works are devoted to the methods for its solution. This is explained not only by theoretical interest to this problem ([1–3]) but also by the existence of numerous applied problems connected with the necessity of solution of linear matrix equation (see also [4–6]). It may be noted that linear matrix equations have application in theory of differential equations, system theory, control theory, stability theory, and some other areas of pure and applied mathematics.

Many authors addressed the question when (1) (over the field of real numbers or the field of complex numbers) has a solution which belongs to a special class of matrices. They are given necessary and sufficient conditions (using generalized inverses) for the existence of symmetric ([7–10]), symmetric with prescribed rank [11], Hermitian and skew-Hermitian ([12, 13]), reflexive and antireflexive [14], and general solutions which are described in terms of original elements or operators. Methods for constructed symmetric solutions of linear matrix equation are proposed in [9, 13, 15, 16]. More details on this problem and many references to original literature can be found in [6, 17–21].

Let be a field, where is or . It is well known (see [7–9]) that (1) over has a symmetric solution if and only if the equation has a solution and . In this article we affirm that this criterion is true for (1) over a principal ideal domain.

The paper is organized as follows. In Section 2 we prove preparatory results of this article. Necessary and sufficient conditions, under which matrix equation over a principal ideal domain has a symmetric solution, are proposed in Section 3. The method used for constructing of symmetric solutions of equation over a principal ideal domain is presented in Section 4. In this section numerical examples are also given.

We note that obtained results are true for the matrix equation , where and are -by- matrices over an elementary divisor domain.

2. Preparatory Results

It is well known that every matrix with entries in a principal ideal domain admits diagonal reduction; that is, for the matrix there exist matrices and such that is a diagonal matrix, where for all and (divides) for all . The elements are called invariant factors of and the diagonal matrix is called the Smith normal form of matrix (see [1], Chapter 15).

Consider the following matrix equation: where , , and is unknown element in . We denote by the augmented matrix of (3); this is the matrix obtained from by adding matrix . The problem of solvability of (3) over a principal ideal domain has a long history (see, e.g., [22–25] and references therein). In more general form, this theory is presented in [1, 3, 26, 27].

Thus, matrix equation over a principal ideal domain is solvable if and only if where are invariant factors of the matrix (see [26], Chapter 1).

In [25] the following statement was proved. Matrix equation (3) over a principal ideal domain is solvable if and only if the Hermite normal forms of the matrices and coincide; that is, matrices and are right equivalent.

In this part we present new necessary and sufficient conditions for the solvability of the equation over a principal ideal domain.

Lemma 1. Let , , , and let be the Smith normal form of matrix , where and . The matrix equation is solvable over if and only if and the matrix admits the representation If matrix equation is solvable, then for arbitrary matrix matrix is a general solution of the equation .

Proof. Let be the solution of the equation and matrices and such that is the Smith normal form of the matrix . From equality it follows thatPut , , and , where and We rewrite equality (9) in the form From this we have and .
Conversely, let and be invertible matrices such that where and , where and . So, we have Let be an arbitrary matrix. It is obvious that next equality holds ThusFrom the above it follows that the matrix is a general solution of equation . This completes the proof.

Let be a field. From Lemma 1 it follows.

Corollary 2. Let and be nonzero matrices and let and be nonsingular matrices such that The equation is solvable over if and only if , where . If equation is solvable, then for arbitrary matrix the matrix is a general solution of equation

3. The Main Result

In this section we establish conditions under which a symmetric solution over a principal ideal domain exists for matrix equation .

Lemma 3. Let and . The homogeneous equation has a nontrivial symmetric solution.

Proof. For the matrix there exist matrices and such that is the Smith normal form of . Let be a nonzero symmetric matrix. For the symmetric matrix we have and the proof of Lemma 3 is complete.

The main result of this article is the following theorem.

Theorem 4. Let . The matrix equation has a symmetric solution over a principal ideal domain if and only if the equation is solvable over and .

Proof.  
Necessity. Let be a symmetric solution of matrix equation ; that is, and . Thus, . From the last equality we have The necessity is proved.
Sufficiency. Let be a solution of equation ; that is, and . This implies that . Further, let . For the matrix there exist matrices and such that is the Smith normal form of . Let us introduce for the following notation .
From equalities and we havePut and . Thus, and . We rewrite system (17) asWe write matrices and in block forms where , , , , , and . From (18) we obtain where
By Lemma 1, from (20) it follows that and . Similarly, from equality (21) it follows that and . Thus, From equality we have . Hence, From the last two equalities we have This implies that . Thus, matrix is symmetric.
It is easily verified that the symmetric matrix is the common solution of equations and . So, for equation there exists a symmetric solution over .
Now we prove that for arbitrary symmetric matrix the matrix is a symmetric solution of equation . By Lemma 3, the symmetric matrix is the solution of the homogenous equation . Put . It is easy to see that Thus, for arbitrary symmetric matrix the matrix is a general symmetric solution of matrix equations . The proof of Theorem 4 is complete.

4. The Representation of Symmetric Solutions

In this section we give new conditions under which for the equation there exists a symmetric solution. If a symmetric solution exists we propose the method for its construction.

Theorem 5. Let and and let and such that is the Smith normal form of matrix . The equation has a symmetric solution if and only if where is a symmetric matrix and .
If a symmetric solution of equation exists, then for arbitrary symmetric matrix the matrix is a general solution of .

Proof.  
Necessity. Let the symmetric matrix be the solution of equation . Further, let and such that where . By Lemma 1, from equality we have where . Put . Multiplying the last equality from the right by , we obtain We write matrices and in block forms where , , and
It is obvious that is a symmetric matrix. Thus, and are symmetric matrices and . From equalitywe obtain that ; that is, is the symmetric matrix and and . It is easily verified that for arbitrary symmetric matrix the following equality holds: The necessity is proved.
Sufficiency. Let and and let be the Smith normal form of matrix , where . Further, let where is symmetric matrix and . For the symmetric matrix we haveFrom equality (40) it follows thatSince is the symmetric matrix we finally obtain that is also the symmetric matrix. From equality (41) we have that . Giving the above, equation has a symmetric solution.
Let be a symmetric matrix. By Lemma 3, matrix is the common symmetric solution of the homogeneous matrix equations and . Hence, the matrix is the general symmetric solution of the equation and the proof is complete.

Consider the following example.

Example 6. Let be the ring of integers and and . Consider the matrix equation For invertible matricesover we have Thus, equation is solvable. Moreover, matrix may be represented in the form where is a symmetric matrix. Thus, equation has a symmetric solution.
For arbitrary the matrix is a symmetric solution of the equation , and the matrix is the general symmetric solution of the matrix equation .

Let be a field. Using this terminology, we have the following corollary to Theorem 5 (see also [7–9, 28]).

Corollary 7. Let and let and such that , where . The equation has a symmetric solution if and only if , where is a symmetric matrix and .
If a symmetric solution of the equation exists, then for arbitrary symmetric matrix the matrix is a general solution of .

Consider the following example.

Example 8. Let be the field of rational numbers and Consider the matrix equation For invertible matrices over we have Thus, equation is solvable.
It is easily verified that . This means that equation has a symmetric solution over and is the general symmetric solution of matrix equation for arbitrary symmetric matrix .

Open Question. Let and be -by- matrices over a commutative ring with an identity element. Further, let the matrix equation be solvable over and the matrix be symmetric. When are these conditions sufficient for the existence of symmetric solution of the equation over ?

Conflicts of Interest

The author declares that he has no conflicts of interest.

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