×

A system of matrix equations and its applications. (English) Zbl 1291.15043

For the following system of matrix equations, \(A_1X = {C_1}\), \({A_2}Y = {C_2}\), \(Y{B_2} = {D_2}\), \(Y = {Y^ * }\), \({A_3}Z = {C_3}\), \(Z{B_3} = {D_3}\), \(Z = {Z^ * }\), \({B_4}X + {({B_4}X)^ * } + {C_4}YC_4^ * + {D_4}ZD_4^ * = {A_4}\), solvability conditions are proved, a general solution is formulated, and the maximal and minimal ranks and inertias of \(Y\) and \(Z\) are established. Finally, for the system \({A_2}Y = {C_2}\), \(Y{B_2} = {D_2}\), \({A_3}Z = {C_3}\), \(Z{B_3} = {D_3}\), \({C_4}YC_4^ * + {D_4}ZD_4^ * = {A_4}\), the maximal and minimal ranks and inertias of general Hermitian solutions are established and some necessary and sufficient conditions to have nonnegative definite, nonpositive definite, positive definite and negative definite solutions are proved.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Baksalary, J K, Nonnegative definite and positive definite solutions to the matrix equation AXA* = \(B\), Linear Multilinear Algebra, 16, 133-139, (1984) · Zbl 0552.15009
[2] Braden, HW, Theequation\(A\)\^{}{T}\(X\) ± \(X\)\^{}{T}\(A\) = \(B\), SIAM J Matrix Anal Appl, 20, 295-302, (1998) · Zbl 0920.15005
[3] Chu, D L; Chan, H; Ho, D W C, Regularrization of singular systems by derivative and proportional output feedback, SIAM J Matrix Anal Appl, 19, 21-38, (1998) · Zbl 0912.93027
[4] Chu, D L; Hung, Y S; Woerdeman, H J, Inertia and rank characterizations of some matrix expressions, SIAM J Matrix Anal Appl, 31, 1187-1226, (2009) · Zbl 1198.15010
[5] Chu, D L; Mehrmann, V; Nichols, N K, Minimum norm regularization of descriptor systems by mixed output feedback, Linear Algebra Appl, 296, 39-77, (1999) · Zbl 0959.93032
[6] Cvetković-IIić, D S; Dajić, A; Koliha, J J, Positive and real-positive solutions to the equation axa* = \(c\) in \(C\)*-algebras, Linear Multilinear Algebra, 55, 535-543, (2007) · Zbl 1180.47014
[7] Cvetković-Ilić, D S, Re-nnd solutions of the matrix equation AXB = \(C\), J Aust Math Soc, 84, 63-72, (2008) · Zbl 1157.15012
[8] Dajić, A; Koliha, J J, Positive solutions to the equations AX = \(C\) and XB = \(D\) for Hilbert space operators, J Math Anal Appl, 333, 567-576, (2007) · Zbl 1120.47009
[9] Dehghan, M; Hajarian, M, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra Appl, 432, 1531-1552, (2010) · Zbl 1187.65042
[10] Dehghan, M; Hajarian, M, Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations, Appl Math Lett, 24, 444-449, (2011) · Zbl 1206.65144
[11] Deng, Y B; Hu, X Y, On solutions of matrix equation AXA\^{}{T} + BY B\^{}{T} = \(C\), J Comput Math, 23, 17-26, (2005) · Zbl 1067.15008
[12] Djordjević, D S, Explicit solution of the operator equation A-X±X-A = \(B\), J Comput Appl Math, 200, 701-704, (2007) · Zbl 1113.47011
[13] Dong, C Z; Wang, Q W; Zhang, Y P, The common positive solution to adjointable operators equations with an application, J Math Anal Appl, 396, 670-679, (2012) · Zbl 1264.47021
[14] Farid, F O; Moslehian, M S; Wang, Q W; etal., On the Hermitian solutions to a system of adjointable operator equations, Linear Algebra Appl, 437, 1854-1891, (2012) · Zbl 1276.47018
[15] Größ, J, A note on the general Hermitian solution to AXA* = \(B\), Bull Malaysian Math Soc, 21, 57-62, (1998) · Zbl 1006.15011
[16] Größ, J, Nonnegative-definite and positive-definite solutions to the matrix equation AXA* = \(B\) revisited, Linear Algebra Appl, 321, 123-129, (2000) · Zbl 0984.15011
[17] He, Z H; Wang, Q W, A real quaternion matrix equation with with applications, Linear Multilinear Algebra, 61, 725-740, (2013) · Zbl 1317.15016
[18] He, Z H; Wang, Q W, Solutions to optimization problems on ranks and inertias of a matrix function with applications, Appl Math Comput, 219, 2989-3001, (2012) · Zbl 1309.15025
[19] 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
[20] Liao, A P; Bai, Z Z, The constrained solutions of two matrix equations, Acta Math Sin English Ser, 18, 671-678, (2002) · Zbl 1028.15011
[21] Liu, Y H; Tian, Y G, A simultaneous decomposition of a matrix triplet with applications, Numer Linear Algebra Appl, 18, 69-85, (2011) · Zbl 1249.15020
[22] Liu, Y H; Tian, Y G, MAX-MIN problems on the ranks and inertias of the matrix expressions \(A\) − BXC ± (BXC)*, J Optim Theory Appl, 148, 593-622, (2011) · Zbl 1223.90077
[23] Liu, Y H; Tian, Y G; Takane, Y, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA* = \(B\), Linear Algebra Appl, 431, 2359-2372, (2009) · Zbl 1180.15018
[24] Marsaglia, G; Styan, G P H, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292, (1974) · Zbl 0297.15003
[25] Piao, F X; Zhang, Q L; Wang, Z F, The solution to matrix equation AX + \(X\)\^{}{T}\(C\) = \(B\), J Franklin Inst, 344, 1056-1062, (2007) · Zbl 1171.15015
[26] Sorensen, D C; Antoulas, A C, The Sylvester equation and approximate balanced reduction, Linear Algebra Appl, 351-352, 671-700, (2002) · Zbl 1023.93012
[27] Tian, Y G, The solvability of two linear matrix equations, Linear Multilinear Algebra, 48, 123-147, (2000) · Zbl 0970.15005
[28] Tian, Y G; Liu, Y H, Extremal ranks of some symmetric matrix expressions with applications, SIAM J Matrix Anal Appl, 28, 890-905, (2006) · Zbl 1123.15001
[29] Tian, Y G, Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method, Nonlinear Anal, 75, 717-734, (2012) · Zbl 1236.65070
[30] Tian, Y G, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A−BX−(BX)* with applications, Linear Algebra Appl, 434, 2109-2139, (2011) · Zbl 1211.15022
[31] Tian Y G. Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions. Arxiv: 1301.2850 · Zbl 1253.15050
[32] Wang, Q W; Chang, H X; Lin, C, On the centro-symmetric solution of a system of matrix equations over a regular ring with identity, Algebra Colloq, 14, 555-570, (2007) · Zbl 1143.15013
[33] Wang, Q W; He, Z H, Some matrix equations with applications, Linear Multilinear Algebra, 60, 1327-1353, (2012) · Zbl 1262.15014
[34] Wang, Q W; Jiang, J, Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation, Electron J Linear Algebra, 20, 552-573, (2010) · Zbl 1207.15016
[35] 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
[36] Wang, Q W; 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
[37] Wang, Q W; Woude, J W; Yu, S W, An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications, Sci China Math, 54, 907-924, (2011) · Zbl 1218.15008
[38] 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
[39] Wang, Q W; Zhang, X; He, Z H, On the Hermitian structures of the solution to a pair of matrix equations, Linear Multilinear Algebra, 61, 73-90, (2012) · Zbl 1264.15020
[40] Wang, Q W; Zhang, X; Woude, J W, A new simultaneous decomposition of a matrix quaternity over an arbitrary division ring with applications, Commun Algebra, 40, 2309-2342, (2012) · Zbl 1252.15014
[41] Wang, Q W; Zhang, H S; Yu, S W, On the real and pure imaginary solutions to the quaternion matrix equation AXB + CY D = \(E\), Electron J Linear Algebra, 17, 343-358, (2008) · Zbl 1154.15019
[42] Wimmer, H K, Consistency of a pair of generalized Sylvester equations, IEEE Trans Automat Control, 39, 1014-1016, (1994) · Zbl 0807.93011
[43] Xu, Q X, Common Hermitian and positive solutions to the adjointable operator equations AX = C,XB = \(D\), Linear Algebra Appl, 429, 1-11, (2008) · Zbl 1153.47012
[44] Xu, Q X; Sheng, L J; Gu, Y Y, The solution to some operator equations, Linear Algebra Appl, 429, 1997-2024, (2008) · Zbl 1147.47014
[45] Yuan, S F; Wang, Q W, Two special kinds of least squares solutions for the quaternion matrix equation AXB+CXD = \(E\), Electron J Linear Algebra, 23, 257-274, (2012) · Zbl 1250.65051
[46] Zhang, X; Wang, Q W; Liu, X, Inertias and ranks of some Hermitian matrix functions with applications, Cent Eur J Math, 10, 329-351, (2012) · Zbl 1253.15050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.