Research Article On the Hermitian R-Conjugate Solution of a System of Matrix Equations

Transcription

1 Applied Mathematics Volume 01, Article ID , 14 pages doi: /01/ Research Article On the Hermitian R-Conjugate Solution of a System of Matrix Equations Chang-Zhou Dong, 1 Qing-Wen Wang, and Yu-Ping Zhang 3 1 School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, Hebei , China Department of Mathematics, Shanghai University, Shanghai, Shanghai 00444, China 3 Department of Mathematics, Ordnance Engineering College, Shijiazhuang, Hebei , China Correspondence should be addressed to Qing-Wen Wang, wqw858@yahoo.com.cn Received 3 October 01; Accepted 6 November 01 Academic Editor: Yang Zhang Copyright q 01 Chang-Zhou Dong et al. his 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. Let R be an n by n nontrivial real symmetric involution matrix, that is, R R 1 R / I n.ann n complex matrix A is termed R-conjugate if A RAR, wherea denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX C and XB D and present an expression of the Hermitian R-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian R-conjugate solution with the least norm to this system mentioned above is considered. he representation of such solution is also derived. Finally, an algorithm and numerical examples are given. 1. Introduction hroughout, we denote the complex m n matrix space by C m n,therealm n matrix space by R m n, and the set of all matrices in R m n with rank r by R m n r. he symbols I,A, A,A,A, and A stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of A C m n, respectively. We use V n to denote the n n backward matrix having the elements 1 along the southwest diagonal and with the remaining elements being zeros. Recall that an n n complex matrix A is centrohermitian if A V n AV n. Centrohermitian matrices and related matrices, such as k-hermitian matrices, Hermitian oeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas see, 1 4. As a generalization of a centrohermitian matrix and related matrices,

2 Applied Mathematics rench 5 gave the definition of R-conjugate matrix. A matrix A C n n is R-conjugate if A RAR, where R is a nontrivial real symmetric involution matrix, that is, R R 1 R and R / I n. At the same time, rench studied the linear equation Az w for R-conjugate matrices in 5, where z, w are known column vectors. Investigating the matrix equation AX B 1.1 with the unknown matrix X being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic As a generalization of 1.1, the classical system of matrix equations AX C, XB D 1. has attracted many author s attention. For instance, 15 gave the necessary and sufficient conditions for the consistency of 1., 16, 17 derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively. Moreover, many results have been obtained about the system 1. with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions see, o our knowledge, so far there has been little investigation of the Hermitian R-conjugate solution to 1.. Motivated by the work mentioned above, we investigate Hermitian R-conjugate solutions to 1.. We also consider the optimal approximation problem X E min X E, X S X 1.3 where E is a given matrix in C n n and S X the set of all Hermitian R-conjugate solutions to 1.. In many cases the system 1. has not Hermitian R-conjugate solution. Hence, we need to further study its least squares solution, which can be described as follows: Let RHC n n denote the set of all Hermitian R-conjugate matrices in C n n : S L { X min AX C XB D )}. X RHC n n 1.4 Find X C n n such that X min X. X S L 1.5 In Section, we present necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to 1. and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to 1.3. InSection4, we provide the least squares Hermitian R-conjugate solution to 1.5. In Section 5, we give an algorithm and a numerical example to illustrate our results.

3 Applied Mathematics 3. R-Conjugate Hermitian Solution to 1. In this section, we establish the solvability conditions and the general expression for the Hermitian R-conjugate solution to 1.. We denote RC n n and RHC n n the set of all R-conjugate matrices and Hermitian R- conjugate matrices, respectively, that is, RC n n HRC n n { } A A RAR, { A A RAR, A A },.1 where R is n n nontrivial real symmetric involution matrix. Chang et al. in 9 mentioned that for nontrivial symmetric involution matrix R R n n, there exist positive integer r and n n real orthogonal matrix P, Q such that R [ P, Q ] I r 0 P,. 0 I n r Q where P R n r,q R n n r.by., RP P, RQ Q, P P I r, Q Q I n r, P Q 0, Q P 0..3 hroughout this paper, we always assume that the nontrivial symmetric involution matrix R is fixed which is given by. and.3. Now, we give a criterion of judging a matrix is R-conjugate Hermitian matrix. heorem.1. AmatrixK HRC n n if and only if there exists a symmetric matrix H R n n such that K ΓHΓ,where Γ [ P, iq ],.4 with P, Q being the same as.. Proof. If K HRC n n, then K RKR. By., K RKR [ P, Q ] I r 0 P K [ P, Q ] I r 0 P,.5 0 I n r Q 0 I n r Q

4 4 Applied Mathematics which is equivalent to P K [ P, Q ] Q I r 0 P K [ P, Q ] I r 0. 0 I n r Q 0 I n r.6 Suppose that P K [ P, Q ] K 11 K 1..7 Q K 1 K Substituting.7 into.6, weobtain K 11 K 1 I r 0 K 11 K 1 I r 0 K 11 K 1..8 K 1 K 0 I n r K 1 K 0 I n r K 1 K Hence, K 11 K 11, K 1 K 1, K 1 K 1, K K, that is, K 11,iK 1,iK 1,K are real matrices. If we denote M ik 1,N ik 1, then by.7 K [ P, Q ] K 11 K 1 P [ P, iq ] K 11 M P..9 K 1 K Q N K iq Let Γ P, iq,and H K 11 M..10 N K hen, K can be expressed as ΓHΓ, where Γ is unitary matrix and H is a real matrix. By K K ΓH Γ K K ΓHΓ,.11 we obtain H H.

5 Applied Mathematics 5 Conversely, if there exists a symmetric matrix H R n n such that K ΓHΓ, then it follows from.3 that RKR RΓHΓ R R [ P, iq ] H P R [ P, iq ] H P ΓHΓ K, iq iq.1 K ΓH Γ ΓHΓ K, that is, K HRC n n. heorem.1 implies that an arbitrary complex Hermitian R-conjugate matrix is equivalent to a real symmetric matrix. Lemma.. For any matrix A C m n,a A 1 ia,where A 1 A A, A A A..13 i Proof. For any matrix A C m n, it is obvious that A A 1 ia, where A 1,A are defined as.13. Now, we prove that the decomposition A A 1 ia is unique. If there exist B 1,B such that A B 1 ib, then A 1 B 1 i B A It follows from A 1,A,B 1, and B are real matrix that A 1 B 1, A B..15 Hence, A A 1 ia holds, where A 1,A are defined as.13. By heorem.1, forx HRC n n, we may assume that X ΓYΓ,.16 where Γ is defined as.4 and Y R n n is a symmetric matrix. Suppose that AΓ A 1 ia C m n,cγ C 1 ic C m n, Γ B B 1 ib C n l,and Γ D D 1 id C n l, where A 1 AΓ AΓ, A B 1 Γ B Γ B AΓ AΓ i, B Γ B Γ B i, C 1 CΓ CΓ, C, D 1 Γ D Γ D CΓ CΓ, i, D Γ D Γ D. i.17

6 6 Applied Mathematics hen, system 1. can be reduced into A 1 ia Y C 1 ic, Y B 1 ib D 1 id,.18 which implies that A 1 Y C 1, Y [ ] [ ] B 1, B D1, D..19 A C Let F A 1, G C 1, K [ ] B 1, B, A L [ D 1, D ], C M F, N G. K L.0 hen, system 1. has a solution X in HRC n n if and only if the real system MY N.1 has a symmetric solution Y in R n n. Lemma.3 heorem 1 in 7. Let A R m n.hesv D of matrix A is as follows A U Σ 0 V, 0 0. where U U 1,U R m m and V V 1,V R n n are orthogonal matrices, Σ diag σ 1,..., σ r,σ i > 0 i 1,...,r, r rank A, U 1 R m r,v 1 R n r. hen, 1.1 has a symmetric solution if and only if AB BA, U B 0..3 In that case, it has the general solution X V 1 Σ 1 U 1 B V V B U 1 Σ 1 V 1 V GV,.4 where G is an arbitrary n r n r symmetric matrix. By Lemma.3, we have the following theorem.

7 Applied Mathematics 7 heorem.4. Given A C m n,c C m n,b C n l, and D C n l. Let A 1,A,C 1, C, B 1,B,D 1,D,F,G,K,L,M,andNbe defined in.17,.0, respectively. Assume that the SV D of M R m l n is as follows M U M 1 0 V, where U U 1,U R m l m l and V V 1,V R n n are orthogonal matrices, M 1 diag σ 1,...,σ r,σ i > 0 i 1,...,r, r rank M, U 1 R m l r,v 1 R n r. hen, system 1. has a solution in HRC n n if and only if MN NM, U N 0..6 In that case, it has the general solution ) X Γ V 1 M 1 1 U 1 N V V N U 1 M 1 1 V 1 V GV Γ,.7 where G is an arbitrary n r n r symmetric matrix. 3. he Solution of Optimal Approximation Problem 1.3 When the set S X of all Hermitian R-conjugate solution to 1. is nonempty, it is easy to verify S X is a closed set. herefore, the optimal approximation problem 1.3 has a unique solution by 30. heorem 3.1. Given A C m n,c C m n,b C n l,d C n l,e C n n, and E 1 1/ Γ EΓ Γ EΓ. Assume S X is nonempty, then the optimal approximation problem 1.3 has a unique solution X and ) X Γ V 1 M 1 1 U 1 N V V N U 1 M 1 1 V 1 V V E 1V V Γ. 3.1 Proof. Since S X is nonempty, X S X has the form of.7. By Lemma., Γ EΓ can be written as Γ EΓ E 1 ie, 3. where E 1 1 ) Γ EΓ Γ EΓ, E 1 ) Γ EΓ Γ i EΓ. 3.3

8 8 Applied Mathematics According to 3. and the unitary invariance of Frobenius norm ) X E Γ V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV Γ E ) 3.4 E 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV. ie We get X E E 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV E. 3.5 hen, min X SX X E is consistent if and only if there exists G R n r n r such that mine 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV. 3.6 For the orthogonal matrix V E 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV V E 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV V ) V 1 E 1 V 1 M 1 U 1 N V V 1 1 E 1 V 1 M 1 U 1 N V V ) E 1 V V N U 1 M 1 V 1 V 1 V ) E 1 V GV. V 3.7 herefore, mine 1 V 1 M 1 U 1 N V V N U 1 M 1 V 1 V GV 3.8 is equivalent to G V E 1V. 3.9 Substituting 3.9 into.7,weobtain he Solution of Problem 1.5 In this section, we give the explicit expression of the solution to 1.5. heorem 4.1. Given A C m n,c C m n,b C n l, and D C n l.leta 1,A,C 1,C, B 1,B,D 1,D,F,G,K,L,M, and N be defined in.17,.0, respectively. Assume that the

9 Applied Mathematics 9 SV D of M R m l n is as.5 and system 1. has not a solution in HRC n n. hen, X S L can be expressed as X ΓV M 1 1 U 1 NV 1 M 1 V N U 1 M 1 1 Y 1 U 1 NV V Γ, 4.1 where Y R n r n r is an arbitrary symmetric matrix. Proof. It yields from.17.1 and.5 that AX C XB D MY N U M 1 0 V Y N 0 0 M 1 0 V YV U NV Assume that V YV Y 11 Y 1, Y 11 R r r, Y R n r n r. 4.3 Y 1 Y hen, we have AX C XB D M 1 Y 11 U 1 NV 1 M 1 Y 1 U 1 NV U NV 1 U NV. 4.4 Hence, min AX C XB D ) 4.5 is solvable if and only if there exist Y 11,Y 1 such that M 1 Y 11 U 1 NV 1 min, M 1 Y 1 U 1 NV min. 4.6

10 10 Applied Mathematics It follows from 4.6 that Y 11 M 1 1 U 1 NV 1, Y 1 M 1 1 U 1 NV. 4.7 Substituting 4.7 into 4.3 and then into.16, we can get that the form of elements in S L is 4.1. heorem 4.. Assume that the notations and conditions are the same as heorem 4.1. hen, X min X S L X 4.8 if and only if X ΓV M 1 1 U 1 NV 1 M 1 V N U 1 M U 1 NV V Γ. 4.9 Proof. In heorem 4.1, it implies from 4.1 that min X SL X is equivalent to X has the expression 4.1 with Y 0. Hence, 4.9 holds. 5. An Algorithm and Numerical Example Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem 1.3 and the least squares problem with least norm 1.5. All the tests are performed by MALAB 6.5 which has a machine precision of around Algorithm Input A C m n,c C m n,b C n l,d C n l. Compute A 1,A,C 1,C,B 1,B,D 1,D,F,G,K,L,M,and N by.17 and.0. 3 Compute the singular value decomposition of M with the form of.5. 4 If.6 holds, then input E C n n and compute the solution X of problem 1.3 according 3.1, else compute the solution X to problem 1.5 by 4.9. o show our algorithm is feasible, we give two numerical example. Let an nontrivial symmetric involution be R

11 Applied Mathematics 11 We obtain P, Q in. by using the spectral decomposition of R, then by i 0 Γ i Example 5.. Suppose A C 4,C C 4,B C 4 3,D C 4 3,and i 45i 7.1 i A, i i i i i 1.801i C, i i 4 1i.369i i 4i 4.66i 8.1 5i B, i 56 i.i i i i i i i i D i i i i i i 5.3 We can verify that.6 holds. Hence, system 1. has an Hermitian R-conjugate solution. Given 7.35i 8.389i i 4.6i i i i E i i i 0 5.1i 0

12 1 Applied Mathematics Applying Algorithm 5.1, we obtain the following: i i i i X i i 0.000i i Example 5. illustrates that we can solve the optimal approximation problem with Algorithm 5.1 when system 1. have Hermitian R-conjugate solutions. Example 5.3. Let A, B, and C be the same as Example 5.,andletD in Example 5. be changed into i i i i i i D i i i i i i We can verify that.6 does not hold. By Algorithm 5.1, weget i i.417i i.83 X i i i i Example 5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system 1. has not Hermitian R-conjugate solutions. Acknowledgments his research was supported by the Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission 13ZZ080, the National Natural Science Foundation of China , the Natural Science Foundation of Shanghai 11ZR141500, the Ph.D. Programs Foundation of Ministry of Education of China , the Discipline Project at the corresponding level of Shanghai A , Shanghai Leading Academic Discipline Project J50101, the Natural Science Foundation of Hebei province A , and the Natural Science Foundation of Hebei province A he authors are grateful to the anonymous referees for their helpful comments and constructive suggestions.

13 Applied Mathematics 13 References 1 R. D. Hill, R. G. Bates, and S. R. Waters, On centro-hermitian matrices, SIAM Journal on Matrix Analysis and Applications, vol. 11, no. 1, pp , R. Kouassi, P. Gouton, and M. Paindavoine, Approximation of the Karhunen-Loeve tranformation and Its application to colour images, Signal Processing: Image Communication, vol. 16, pp , A. Lee, Centro-Hermitian and skew-centro-hermitian matrices, Linear Algebra and Its Applications, vol. 9, pp , Z.-Y. Liu, H.-D. Cao, and H.-J. Chen, A note on computing matrix-vector products with generalized centrosymmetric centrohermitian matrices, Applied Mathematics and Computation, vol. 169, no., pp , W. F. rench, Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra and Its Applications, vol. 377, pp , K.-W. E. Chu, Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra and Its Applications, vol. 119, pp , H. Dai, On the symmetric solutions of linear matrix equations, Linear Algebra and Its Applications, vol. 131, pp. 1 7, F. J. H. Don, On the symmetric solutions of a linear matrix equation, Linear Algebra and Its Applications, vol. 93, pp. 1 7, I. Kyrchei, Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations, Linear Algebra and Its Applications, vol. 438, no. 1, pp , Y. Li, Y. Gao, and W. Guo, A Hermitian least squares solution of the matrix equation AXB C subject to inequality restrictions, Computers & Mathematics with Applications, vol. 64, no. 6, pp , Z.-Y. Peng and X.-Y. Hu, he reflexive and anti-reflexive solutions of the matrix equation AX B, Linear Algebra and Its Applications, vol. 375, pp , L. Wu, he re-positive definite solutions to the matrix inverse problem AX B, Linear Algebra and Its Applications, vol. 174, pp , S. F. Yuan, Q. W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB CYD E over different constrained matrices, Article ID 766, International Computer Mathematics. In press. 14 Z.-Z. Zhang, X.-Y. Hu, and L. Zhang, On the Hermitian-generalized Hamiltonian solutions of linear matrix equations, SIAM Journal on Matrix Analysis and Applications, vol. 7, no. 1, pp , F. Cecioni, Sopra operazioni algebriche, Annali della Scuola Normale Superiore di Pisa, vol. 11, pp. 17 0, K.-W. E. Chu, Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra and Its Applications, vol , pp , S. K. Mitra, A pair of simultaneous linear matrix equations A 1 XB 1 C 1,A XB C and a matrix programming problem, Linear Algebra and Its Applications, vol. 131, pp , H.-X. Chang and Q.-W. Wang, Reflexive solution to a system of matrix equations, Shanghai University, vol. 11, no. 4, pp , A. Dajić and J. J. Koliha, Equations ax c and xb d in rings and rings with involution with applications to Hilbert space operators, Linear Algebra and Its Applications, vol. 49, no. 7, pp , A. Dajić and J. J. Koliha, Positive solutions to the equations AX C and XB D for Hilbert space operators, Mathematical Analysis and Applications, vol. 333, no., pp , C.-Z. Dong, Q.-W. Wang, and Y.-P. Zhang, he common positive solution to adjointable operator equations with an application, Mathematical Analysis and Applications, vol. 396, no., pp , 01. X. Fang, J. Yu, and H. Yao, Solutions to operator equations on Hilbert C -modules, Linear Algebra and Its Applications, vol. 431, no. 11, pp , C. G. Khatri and S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM Journal on Applied Mathematics, vol. 31, no. 4, pp , I. Kyrchei, Analogs of Cramer s rule for the minimum norm least squares solutions of some matrix equations, Applied Mathematics and Computation, vol. 18, no. 11, pp , F. L. Li, X. Y. Hu, and L. Zhang, he generalized reflexive solution for a class of matrix equations AX C, XB D, Acta Mathematica Scientia, vol. 8B, no. 1, pp , 008.

14 14 Applied Mathematics 6 Q.-W. Wang and C.-Z. Dong, Positive solutions to a system of adjointable operator equations over Hilbert C -modules, Linear Algebra and Its Applications, vol. 433, no. 7, pp , Q. Xu, Common Hermitian and positive solutions to the adjointable operator equations AX C, XB D, Linear Algebra and Its Applications, vol. 49, no. 1, pp. 1 11, F. Zhang, Y. Li, and J. Zhao, Common Hermitian least squares solutions of matrix equations A 1 XA 1 B 1 and A XA B subject to inequality restrictions, Computers & Mathematics with Applications, vol. 6, no. 6, pp , H.-X. Chang, Q.-W. Wang, and G.-J. Song, R, S -conjugate solution to a pair of linear matrix equations, Applied Mathematics and Computation, vol. 17, no. 1, pp. 73 8, E. W. Cheney, Introduction to Approximation heory, McGraw-Hill, New York, NY, USA, 1966.

15 Advances in Operations Research Advances in Decision Sciences Mathematical Problems in Engineering Algebra Probability and Statistics he Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Stochastic Analysis Abstract and Applied Analysis International Mathematics Discrete Dynamics in Nature and Society Discrete Mathematics Applied Mathematics Function Spaces Optimization