Journal of Applied Mathematics

Volume 2012 (2012), Article ID 471573, 19 pages

doi:10.1155/2012/471573

Research Article

Constrained Solutions of a System of Matrix Equations

Qing-Wen Wang¹ and Juan Yu1,2

¹Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China

²Department of Basic Mathematics, China University of Petroleum, Qingdao 266580, China

Received 26 September 2012; Accepted 7 December 2012

Academic Editor: Panayiotis J.  Psarrakos

Copyright © 2012 Qing-Wen Wang and Juan Yu. 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 derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equations and , respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

1. Introduction

Throughout this paper, the following notations will be used. , , , and denote the set of all real matrices, the set of all orthogonal matrices, the set of all symmetric matrices, and the set of all skew-symmetric matrices, respectively. is the identity matrix of order . and represent the transpose and the trace of the real matrix, respectively. stands for the Frobenius norm induced by the inner product. The following two definitions will also be used.

Definition 1.1 (see [1]). A real matrix is said to be a symmetric orthogonal matrix if and .

Definition 1.2 (see [2]). A real matrix is called a skew-symmetric orthogonal matrix if and .

The set of all symmetric orthogonal matrices and the set of all skew-symmetric orthogonal matrices are, respectively, denoted by and . Since the linear matrix equation(s) and its optimal approximation problem have great applications in structural design, biology, control theory, and linear optimal control, and so forth, see, for example, [3–5], there has been much attention paid to the linear matrix equation(s). The well-known system of matrix equations

as one kind of linear matrix equations, has been investigated by many authors, and a series of important and useful results have been obtained. For instance, the system (1.1) with unknown matrix being bisymmetric, centrosymmetric, bisymmetric nonnegative definite, Hermitian and nonnegative definite, and -(skew) symmetric has been, respectively, investigated by Wang et al. [6, 7], Khatri and Mitra [8], and Zhang and Wang [9]. Of course, if the solvability conditions of system (1.1) are not satisfied, we may consider its least squares solution. For example, Li et al. [10] presented the least squares mirrorsymmetric solution. Yuan [11] got the least-squares solution. Some results concerning the system (1.1) can also be found in [12–18].

Symmetric orthogonal matrices and skew-symmetric orthogonal matrices play important roles in numerical analysis and numerical solutions of partial differential equations. Papers [1, 2], respectively, derived the symmetric orthogonal solution of the matrix equation and the skew-symmetric orthogonal solution of the matrix equation . Motivated by the work mentioned above, we in this paper will, respectively, study the orthogonal solutions, symmetric orthogonal solutions, and skew-symmetric orthogonal solutions of the system (1.1). Furthermore, if the solvability conditions are not satisfied, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) will be also given.

The remainder of this paper is arranged as follows. In Section 2, some lemmas are provided to give the main results of this paper. In Sections 3, 4, and 5, the necessary and sufficient conditions of and the expression for the orthogonal, the symmetric orthogonal, and the skew-symmetric orthogonal solutions of the system (1.1) are, respectively, obtained. In Section 6, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) are presented, respectively. In addition, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable. Finally, in Section 7, some concluding remarks are given.

2. Preliminaries

In this section, we will recall some lemmas and the special - decomposition which will be used to get the main results of this paper.

Lemma 2.1 (see [1, Lemmas 1 and 2]). Given ,  . The matrix equation has a solution if and only if . Let the singular value decompositions of and be, respectively,

where

Then the orthogonal solutions of can be described as

where is arbitrary.

Lemma 2.2 (see [2, Lemmas and ]). Given ,  . The matrix equation has a solution if and only if . Let the singular value decompositions of and be, respectively,

where

Then the orthogonal solutions of can be described as

where is arbitrary.

Lemma 2.3 (see [2, Theorem 1]). If

then the - decomposition of can be expressed as

where ,

Lemma 2.4 (see [1, Theorem ]). If

then the - decomposition of can be described as

where ,  ,  ,

Remarks 2.5. In order to know the - decomposition of an orthogonal matrix with a leading (skew-) symmetric submatrix for details, one can deeply study the proof of Theorem  1 in [1] and [2].

Lemma 2.6. Given ,  . Then the matrix equation has a solution if and only if and . When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as

where

and is arbitrary.

Proof. The Necessity. Assume is a solution of the matrix equation , then we have

The Sufficiency. Since the equality holds, then by Lemma 2.2, the singular value decompositions of and can be, respectively, expressed as (2.4). Moreover, the condition means

which can be written as

From Lemma 2.2, the orthogonal solutions of the matrix equation can be described as (2.6). Now we aim to find that in (2.6) is also symmetric. Suppose that is symmetric, then we have

together with the partitions of the matrices and in Lemma 2.2, we get

By (2.17), we can get (2.19). Now we aim to find the orthogonal solutions of the system of matrix equations (2.20) and (2.21). Firstly, we obtain from (2.20) that , then by Lemma 2.2, (2.20) has an orthogonal solution . By (2.17), the leading principal submatrix of the orthogonal matrix is symmetric. Then we have, from Lemma 2.4,

From (2.20), (2.22), and (2.23), the orthogonal solution of (2.20) is

where is arbitrary. Combining (2.21), (2.24), and (2.25) yields is a symmetric orthogonal matrix. Denote

then the symmetric orthogonal solutions of the matrix equation can be expressed as

Let the partition matrix be

compatible with the block matrix

Put

then the symmetric orthogonal solutions of the matrix equation can be described as (2.13).

Setting ,  , and in [2, Theorem 2], and then by Lemmas 2.1 and 2.3, we can have the following result.

Lemma 2.7. Given ,  . Then the equation has a solution if and only if and . When these conditions are satisfied, the skew-symmetric orthogonal solutions of the matrix equation can be described as

where

and is arbitrary.

3. The Orthogonal Solutions of the System (1.1)

The following theorems give the orthogonal solutions of the system (1.1).

Theorem 3.1. Given ,   and ,  , suppose the singular value decompositions of and are, respectively, as (2.4). Denote

where , , and . Let the singular value decompositions of and be, respectively,

where ,  ,  ,   is diagonal, whose diagonal elements are nonzero singular values of or . Then the system (1.1) has orthogonal solutions if and only if

In which case, the orthogonal solutions can be expressed as

where

and is arbitrary.

Proof. Let the singular value decompositions of and be, respectively, as (2.4). Since the matrix equation has orthogonal solutions if and only if

then by Lemma 2.2, its orthogonal solutions can be expressed as (2.6). Substituting (2.6) and (3.1) into the matrix equation , we have and . By Lemma 2.1, the matrix equation has orthogonal solution if and only if

Let the singular value decompositions of and be, respectively,

where ,  ,  ,   is diagonal, whose diagonal elements are nonzero singular values of or . Then the orthogonal solutions can be described as

where is arbitrary. Therefore, the common orthogonal solutions of the system (1.1) can be expressed as

where

and is arbitrary.

The following theorem can be shown similarly.

Theorem 3.2. Given ,   and ,  , let the singular value decompositions of and be, respectively, as (2.1). Partition

where ,  . Assume the singular value decompositions of and are, respectively,

where ,  ,  ,  is diagonal, whose diagonal elements are nonzero singular values of or . Then the system (1.1) has orthogonal solutions if and only if

In which case, the orthogonal solutions can be expressed as

where

and is arbitrary.

4. The Symmetric Orthogonal Solutions of the System (1.1)

We now present the symmetric orthogonal solutions of the system (1.1).

Theorem 4.1. Given ,  . Let the symmetric orthogonal solutions of the matrix equation be described as in Lemma 2.6. Partition

where ,  . Then the system (1.1) has symmetric orthogonal solutions if and only if

In which case, the solutions can be expressed as

where

and is .

Proof. From Lemma 2.6, we obtain that the matrix equation has symmetric orthogonal solutions if and only if and . When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as

where is arbitrary, ,  . Inserting (4.1) and (4.5) into the matrix equation , we get and . By [1, Theorem  2], the matrix equation has a symmetric orthogonal solution if and only if

In which case, the solutions can be described as

where is arbitrary, , and . Hence the system (1.1) has symmetric orthogonal solutions if and only if all equalities in (4.2) hold. In which case, the solutions can be expressed as

that is, the expression in (4.3).

The following theorem can also be obtained by the method used in the proof of Theorem 4.1.

Theorem 4.2. Given ,  . Let the symmetric orthogonal solutions of the matrix equation be described as

where ,  ,  . Partition

Then the system (1.1) has symmetric orthogonal solutions if and only if

In which case, the solutions can be expressed as

where

and is arbitrary.

5. The Skew-Symmetric Orthogonal Solutions of the System (1.1)

In this section, we show the skew-symmetric orthogonal solutions of the system (1.1).

Theorem 5.1. Given ,  . Suppose the matrix equation has skew-symmetric orthogonal solutions with the form

where is arbitrary, ,  . Partition

where ,  . Then the system (1.1) has skew-symmetric orthogonal solutions if and only if

In which case, the solutions can be expressed as

where

and is arbitrary.

Proof. By [2, Theorem 2], the matrix equation has the skew-symmetric orthogonal solutions if and only if and . When these conditions are satisfied, the general skew-symmetric orthogonal solutions can be expressed as (5.1). Substituting (5.1) and (5.2) into the matrix equation , we get and . From Lemma 2.7, equation has a skew-symmetric orthogonal solution if and only if

When these conditions are satisfied, the solution can be described as

where is arbitrary, ,  . Inserting (5.7) into (5.1) yields that the system (1.1) has skew-symmetric orthogonal solutions if and only if all equalities in (5.3) hold. In which case, the solutions can be expressed as (5.4).

Similarly, the following theorem holds.

Theorem 5.2. Given ,  . Suppose the matrix equation has skew-symmetric orthogonal solutions with the form

where is arbitrary, ,  . Partition

where ,  . Then the system (1.1) has skew-symmetric orthogonal solutions if and only if

In which case, the solutions can be expressed as

where

and is arbitrary.

6. The Least Squares (Skew-) Symmetric Orthogonal Solutions of the System (1.1)

If the solvability conditions of a system of matrix equations are not satisfied, it is natural to consider its least squares solution. In this section, we get the least squares (skew-) symmetric orthogonal solutions of the system (1.1), that is, seek such that

With the help of the definition of the Frobenius norm and the properties of the skew-symmetric orthogonal matrix, we get that

Let

Then, it follows from the skew-symmetric matrix that

Therefore, (6.1) holds if and only if (6.4) reaches its maximum. Now, we pay our attention to find the maximum value of (6.4). Assume the eigenvalue decomposition of is

with

Denote

partitioned according to

then (6.4) has the following form:

Thus, by

where

Equation (6.9) gets its maximum. Since is skew-symmetric, it follows from

where is arbitrary, that (6.1) obtains its minimum. Hence we have the following theorem.

Theorem 6.1. Given and , denote

and let the spectral decomposition of be (6.5). Then the least squares skew-symmetric orthogonal solutions of the system (1.1) can be expressed as (6.12).

If in (6.1) is a symmetric orthogonal matrix, then by the definition of the Frobenius norm and the properties of the symmetric orthogonal matrix, (6.2) holds. Let

Then we get that

Thus (6.15) reaches its minimum if and only if obtains its maximum. Now, we focus on finding the maximum value of . Let the spectral decomposition of the symmetric matrix be

where

Denote

being compatible with

Then

Therefore, it follows from

that (6.20) reaches its maximum. Since is a symmetric orthogonal matrix, then when has the form

where is arbitrary, (6.15) gets its minimum. Thus we obtain the following theorem.

Theorem 6.2. Given ,  , denote

and let the eigenvalue decomposition of be (6.16). Then the least squares symmetric orthogonal solutions of the system (1.1) can be described as (6.22).

Algorithm 6.3. Consider the following.

. Input and .. Compute

. Compute the spectral decomposition of with the form (6.16).. Compute the least squares symmetric orthogonal solutions of (1.1) according to (6.22).

Example 6.4. Assume

It can be verified that the given matrices , and do not satisfy the solvability conditions in Theorem 4.1 or Theorem 4.2. So we intend to derive the least squares symmetric orthogonal solutions of the system (1.1). By Algorithm 6.3, we have the following results:

(1)the least squares symmetric orthogonal solution

(2)

Remark 6.5. (1) There exists a unique symmetric orthogonal solution such that (6.1) holds if and only if the matrix

where

is invertible. Example 6.4 just illustrates it.

(2) The algorithm about computing the least squares skew-symmetric orthogonal solutions of the system (1.1) can be shown similarly; we omit it here.

7. Conclusions

This paper is devoted to giving the solvability conditions of and the expressions of the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions to the system (1.1), respectively, and meanwhile obtaining the least squares symmetric orthogonal and skew-symmetric orthogonal solutions of the system (1.1). In addition, an algorithm and an example have been provided to compute its least squares symmetric orthogonal solutions.

Acknowledgments

This research was supported by the grants from Innovation Program of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), and Shanghai Leading Academic Discipline Project (J50101).

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