## A representation of the general common solution to the matrix equations $$A_1XB_1=C_1$$ and $$A_2XB_2=C_2$$ with applications.(English)Zbl 0983.15016

New necessary and sufficient conditions are derived for a pair of matrix equations $$A_1XB_1=C_1$$ and $$A_2XB_2=C_2$$ to have a common solution. Then a new representation is derived for the general common solution. The result is used to determine conditions for the existence of a solution and a new representation of the general Hermitian solution to the matrix equation $$AXB=C$$ $$(A,B$$, and $$C$$ are known matrices over the complex field).

### MSC:

 15A24 Matrix equations and identities
Full Text:

### References:

 [1] Morris, G.L.; Odell, P.L., Common solutions for n matrix equations with applications, J. assn. com. Mach., 15, 272-274, (1968) · Zbl 0157.22602 [2] Mitra, S.K., Common solutions to a pair of linear matrix equations A1{\bfxb}1 = {\bfc}1 and {\bfa}2{\bfxb}2 = {\bfc}2, Proc. Cambridge philos. soc., 74, 213-216, (1973) [3] Mitra, S.K., A pair of simultaneous linear matrix equations and a matrix programming problem, Linear algebra appl., 131, 97-123, (1990) [4] Shinozaki, N.; Sibuya, M., Consistency of a pair of matrix equations with an application, Keio engrg. rep., 27, 141-146, (1974) [5] von der Woude, J., Feedback decoupling and stabilization for linear systems with multiple exogenous variables, Ph.D. thesis, (1987), Technical Univ. of Eindhoven Netherlands [6] Özgüler, A.B.; Akar, N., A common solution to a pair of linear matrix equations over a principal domain, Linear algebra appl., 144, 85-99, (1991) · Zbl 0718.15006 [7] Jones, J.; Narathong, C., Estimation of variance and covariance components in linear models containing multiparameter matrices, Mathl. comput. modelling, 11, 1097-1100, (1988) [8] Chu, K.W.E., Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear algebra appl., 88/89, 83-98, (1987) · Zbl 0612.15003 [9] von Rosen, D., Some results on homogeneous matrix equations, SIAM J. matrix anal., 14, 137-145, (1993) · Zbl 0768.15008 [10] Bhimasankaram, P., Common solutions to the linear matrix equations AX = {\bfb}, {\bfcx} = {\bfd}, and {\bfexf} = {\bfg}, Sankhya ser. A, 38, 404-409, (1976) · Zbl 0411.15008 [11] Magnus, J.R., L-structured matrices and linear matrix equations, Linear and multilinear algebra, 14, 67-88, (1983) · Zbl 0527.15006 [12] Campbell, S.L.; Meyer, C.D., Generalized inverses of linear transformations, (1979), Dover Publications New York · Zbl 0417.15002 [13] Young, D.M.; Seaman, J.W.; Meaux, L.M., A characterization of independence distribution-preserving covariance structures for the multivariate linear model, J. mult. anal., 68, 165-175, (1999) · Zbl 0927.62057 [14] D.M. Young, A. Navarra and P.L. Odell, A representation of the general common nonnegative-definite solution to a system of linear homogeneous matrix equations, (submitted). · Zbl 0983.15016 [15] Mitra, S.K., A pair of simultaneous linear matrix equations and a matrix programming problem, Linear algebra appl., 131, 107-123, (1990) · Zbl 0712.15010 [16] Khatri, C.G.; Mitra, S.K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. appl. math., 31, 578-585, (1976) · Zbl 0359.65033 [17] Vetter, W.J., Vector structures and solutions of linear matrix equations, Linear algebra appl., 9, 181-188, (1975) · Zbl 0307.15003 [18] Magnus, J.R.; Neudecker, H., The elimination matrix: some lemmas and applications, SIAM J. algebraic discrete methods, 1, 422-428, (1980) · Zbl 0497.15014 [19] Henk Don, F.J., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 83, 1-7, (1987) · Zbl 0622.15001 [20] Hua, D., On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009
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.