The <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M1" height="13.9607pt" version="1.1" viewBox="-0.0657574 -13.8949 8.4665 13.9607" width="8.4665pt"><g transform="matrix(.013,0,0,-0.013,0,-7.899)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g></svg>Congruence Class of the Solutions to a System of Matrix Equations

Zhang, Yu-Ping; Dong, Chang-Zhou

doi:https://doi.org/10.1155/2014/703529

Journal of Applied Mathematics

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Abstract Introduction Acknowledgments References Copyright Related Articles

Research Article | Open Access

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

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

Yu-Ping Zhang¹and Chang-Zhou Dong¹

Academic Editor: Qing-Wen Wang

Received14 May 2014

Revised18 Aug 2014

Accepted20 Aug 2014

Published03 Sept 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 [2–10]). Studying the least-square solutions of the system of matrix equations (1) is also a very active research topic (see [11–16]). 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

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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