Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 703529 | 7 pages | https://doi.org/10.1155/2014/703529

The Congruence Class of the Solutions to a System of Matrix Equations

Academic Editor: Qing-Wen Wang
Received14 May 2014
Revised18 Aug 2014
Accepted20 Aug 2014
Published03 Sep 2014

Abstract

We present the congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation by generalized singular value decomposition and canonical correlation decomposition.

1. Introduction

Throughout we denote the complex matrix space by . The symbols , , and stand for the identity matrix with the appropriate size, the conjugate transpose, and the Frobenius norm of , respectively. Recall that matrices , are in the same congruence class if there is a nonsingular such that [1].

Investigating the classical system of matrix equations has attracted many people’s attention and many results have been obtained about system (1) with various constraints, such as Hermitian, positive definite, positive semidefinite, reflexive, and generalized reflexive solutions (see [210]). Studying the least-square solutions of the system of matrix equations (1) is also a very active research topic (see [1116]). It is well known that Hermitian, positive definite and positive semidefinite matrices are the special case of congruence. Therefore investigating the congruence class of a solution of the matrix equation (1) is very meaningful.

In 2005, Horn et al. [1] studied the possible congruence class of a square solution when linear matrix equation is solvable. In 2009, Zheng et al. [17] describe congruence class of least-square and minimum norm least-square solutions of the equation when it is not solvable and discuss a congruence class of the solutions of the system (1) when it is solvable. To our knowledge, so far there has been little investigation of congruence class of the least-square and minimum norm least-square solutions to (1) when it is not solvable.

Motivated by the work mentioned above, we investigate the congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation (1) by generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD).

2. The Congruence Class of the Solutions to (1)

Lemma 1 (see [4]). Let and . Then the GSVD of and can be expressed as where   and are unitary matrices, is nonsingular matrix, and are identity matrices, and are zero matrices, and with , , and .

For convenience, in the following theorem we denote

Theorem 2. Let , , , , and the GSVD of and be expressed as (2), and then one has the following. (a)The system of matrix equation (1) has a solution in if and only if (b)In that case, the general solutions of (1) are where , , , and are arbitrary. (c)For arbitrary , , , and , there exists a solution in of (1) which is congruent to (d)There exists a minimum norm solution in of (1) which is congruent to

Proof. Using the GSVD of and given by (2), we get By (2) and (5), and have the following matrix decomposition: and we have that system (1) is equivalent to obviously, the system of matrix equation (1) has a solution in if and only if Therefore, (1) has a solution in if and only if (7) holds, and a general form of the solutions can be expressed as (8); for arbitrary , , , and , there exists a solution in of (1) which is congruent to (9), and the part (d) follows from the definition of Frobenius norm.

Remark 3. In 2009, Zheng et al. [17] discuss a congruence class of the solutions of the system (1) when it is solvable. Our result in Theorem 2 is different with the result mentioned above.

3. The Congruence Class of the Least-Square Solutions to (1)

Lemma 4 (see [18]). Let the CCD of matrix pair with , ,  , and   be given as where is a unitary matrix and are nonsingular matrices with the same row partitioning, and ,

Lemma 5 (see [18]). Given , , then there exists a unique matrix such that and can be expressed as

Lemma 6 (see [10]). Given , , , , , and , then there exists a unique matrix such that and can be expressed as where

Using Lemmas 5 and 6, we can easily obtain the following.

Lemma 7. Given , , , , , , , , and , then there exist unique matrices and such that and and can be expressed as where

Lemma 8. Given , , , , , and , then there exist unique matrices and such that and and can be expressed as

Let , , , , and . According to Lemma 4, there exist a unitary matrix and nonsingular matrices and , such that the CCD of matrix pair is given as where , , where , Without loss of generality, let , and then we have the following results.

Theorem 9. Let , , , , and the CCD of matrix pair be expressed as (28), and then one has the following. (a)The least-square solutions to the system (1) arewhere , , , , , and are arbitrary, ,  ,  , and ,  ,  .(b)For arbitrary , , , , , and , there exists a least-square solution in of (1) which is congruent towhere ,  ,  ,  and ,  ,  .(c)There exists a minimum norm least-square solution in of (1) which is congruent towhere ,  ,  ,  and ,  ,  .

Proof. It follows from (28) that Then, Assume that and then By Lemmas 5, 7, and 8, a general form of the least-square solutions can be expressed as (31); for arbitrary , , , , , and , there exists a least-square solution in of (1) which is congruent to (32), and the part (c) follows from the definition of Frobenius norm.

4. An Algorithm and Numerical Examples

Based on the main results of this paper, we in this section propose an algorithm for finding the least-square solutions to the system (1). All the tests are performed by MATLAB 6.5 which has a machine precision of around .

Algorithm 1. (1) Input and , and compute ,  ,  ,  , , and  by the CCD of matrix pair .
(2) Input , , and compute and according to (37).
(3) Compute the least-square solutions of the system (1) by (31).
(4) Compute the congruence class of the least-square and the minimum norm least-square solutions to the system (1) according to (32) and (33).

Example 1. Suppose
Applying Algorithm 1, we obtain the following:
The least-square solutions to the system (1) arewhere , , , , , and are arbitrary.
For arbitrary , , , , , and , there exists a least-square solution in of (1) which is congruent toThere exists a minimum norm least-square solution in of (1) which is ongruent to

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Youth Funds of Natural Science Foundation of Hebei province (A2012403013), the Education Department Foundation of Hebei province (Z2013110), and the Natural Science Foundation of Hebei province (A2012205028).

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Copyright © 2014 Yu-Ping Zhang and Chang-Zhou Dong. 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.

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