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(\(R,S\))-conjugate solution to a pair of linear matrix equations. (English) Zbl 1201.15006
Authors’ abstract: Let \(R\) and \(S\) be \(m \times m\) and \(n \times n\) nontrivial real symmetric involutions. An \(m \times n\) complex matrix \(A\) is termed \((R, S)\)-conjugate if \(\bar A = RAS\), where \(\bar A\) denotes the conjugate of \(A\). In this paper, necessary and sufficient conditions are established for the existence of the \((R, S)\)-conjugate solution to the system of matrix equations \(AX = C\) and \(XB = D\). The expression is also presented for such solution to this system. In addition, the explicit expression of this solution to the corresponding optimal approximation problem is obtained. Furthermore, the least squares \((R, S)\)-conjugate solution with least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally an algorithm and numerical examples are given.

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
Full Text: DOI
[1] Lee, A., Centrohermitian and skew-Centrohermitian matrices, Linear algebra appl., 29, 205-210, (1980) · Zbl 0435.15019
[2] Hill, R.D.; Bates, R.G.; Waters, S.R., On Centrohermitian matrices, SIAM J. matrix anal. appl., 11, 1, 128-133, (1990) · Zbl 0709.15021
[3] Fassbender, H.; Ikramov, K.D., Computing matrix vector products with centrosymmetric and Centrohermitian, Linear algebra appl., 364, 235-241, (2003) · Zbl 1016.65022
[4] Hill, R.D.; Waters, S.R., On K-real and K-Hermitian matrices, Linear algebra appl., 169, 17-29, (1992) · Zbl 0757.15011
[5] Wilkes, D.M.; Morgera, S.D.; Noor, F.; Hayes, M.H., A Hermitian Toeplitz matrix is unitarily similar to a real Toeplitz-plus-Hankel matrix, IEEE trans. signl process., 39, 2146-2148, (1991) · Zbl 0739.65044
[6] Liu, Z.Y.; Cao, H.D.; Chen, H.J., A note computing matrix-vector products with generalized centrosymmetric (Centrohermitian) matrices, Appl. math. comput., 169, 1332-1345, (2005) · Zbl 1103.65049
[7] Oppenheim, A.V., Applications of digital signal processing, (1978), Prentice-Hall Englewood Cliffs
[8] Ng, M.K.; Plemmons, R.J.; Pimentel, F., A new approach to constrained total least squares image restoration, Linear algebra appl., 316, 237-258, (2000) · Zbl 0960.65044
[9] Kouassi, R.; Gouton, P.; Paindavoine, M., Approximation of the karhunen – loeve tranformation and its application to colour images, Signal process.: image commun., 16, 541-551, (2001)
[10] Trench, W.F., Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear algebra appl., 377, 207-218, (2004) · Zbl 1046.15028
[11] Trench, W.F., Characterization and properties of (R,S)-symmetric, (R,S)-skew symmetric, and (R,S)-conjugate matrices, SIAM J. matrix anal. appl., 26, 3, 748-757, (2005) · Zbl 1080.15022
[12] Cecioni, F., Sopra operazioni algebriche, Ann. scuola norm. sup. Pisa sci. fis. mat., 11, 17-20, (1910) · JFM 41.0193.02
[13] Chu, W.E., Singular value and generalized singular value decomposition and the solution of linear matrix equations, Linear algebra appl., 88/89, 83-98, (1987) · Zbl 0612.15003
[14] Mitra, S.K., A pair of simultaneous linear matrix equations A1XB1=C1, A2XB2=C2 and a matrix programming problem, Linear algebra appl., 131, 107-123, (1990)
[15] Khatri, C.G.; Mitra, S.K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. appl. math., 31, 579-585, (1976) · Zbl 0359.65033
[16] Mitra, S.K., The matrix equations AX=C, XB=D, Linear algebra appl., 59, 171-181, (1984) · Zbl 0543.15011
[17] Wang, Q.W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. appl., 49, 641-650, (2005) · Zbl 1138.15003
[18] Wang, Q.W., A system of four matrix equations over von Neumann regular rings and its applications, Acta math. sin., engl. ser., 21, 2, 323-334, (2005) · Zbl 1083.15021
[19] Wang, Q.W., The general solution to a system of real quaternion matrix equations, Comput. math. appl., 49, 665-675, (2005) · Zbl 1138.15004
[20] Wang, Q.W.; Song, G.J.; Liu, X., Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring, Algebra colloq., 16, 2, 293-308, (2009) · Zbl 1176.15020
[21] Wang, Q.W.; Chang, H.X.; Lin, C.Y., P-(skew) symmetric common solutions to a pair of quaternions matrix equations, Appl. math. comput., 195, 721-732, (2008) · Zbl 1149.15011
[22] Wang, Q.W.; Li, C.K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear algebra appl., 430, 1626-1640, (2009) · Zbl 1158.15010
[23] Li, Y.T.; Wu, W.J., Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations, Comput. math. appl., 55, 6, 1142-1147, (2008) · Zbl 1143.15012
[24] Chang, H.X.; Wang, Q.W., Reflexive solution to a system of matrix equations, J. Shanghai univ. (engl. ed.), 11, 4, 355-358, (2007) · Zbl 1141.15319
[25] Li, F.L.; Hu, X.Y.; Zhang, L., The generalized reflexive solution for a calss of matrix equations (AX=C, XB=D), Acta math. scientia, 28B, 1, 185-193, (2008)
[26] Dajic, Alegra; Koliha, J.J., Positive solutions to the equations AX=C, XB=D for Hilbert space operators, J. math. anal. appl., 333, 567-576, (2007) · Zbl 1120.47009
[27] Qiu, Y.Y.; Zhang, Z.Y.; Lu, J.F., The matrix equations AX=B, XC=D with PX=sxp constraint, Appl. math. comput., 189, 1428-1434, (2007) · Zbl 1124.15009
[28] Xu, Q., Common Hermitian and positive solutions to the adjointable operator equations AX=C, XB=D, Linear algebra appl., 429, 1-11, (2008) · Zbl 1153.47012
[29] Wang, Q.W.; van der Woude, J.W.; Chang, H.X., A system of real quaternion matrix equations with applications, Linear algebra appl., 431, 2291-2303, (2009) · Zbl 1180.15019
[30] Wang, Q.W.; Zhou, Y.; Zhang, Q., Ranks of the common solution to six quaternion matrix equations, Acta math. appl. sin. (engl. ser.), (2010)
[31] Wang, Q.W.; Wu, Z.C., Common Hermitian solutions to some operator equations on Hilbert C∗-modules, Linear algebra appl., 432, 3159-3171, (2010) · Zbl 1197.47031
[32] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its application, (1971), Wiley New York
[33] Cheney, E.W., Introduction to approximation theory, (1966), McGraw-Hill Book CO · Zbl 0161.25202
[34] Peng, Z.Y.; Deng, Y.B.; Liu, J.W., Least-squares solution of inverse problem for Hermitian anti-reflexive matrices and its approximation, Acta. math. sin. (engl. ser.), 22, 2, 477-484, (2006) · Zbl 1105.65044
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