A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

AbstractWe consider the system of four linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over ℛ, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X = C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over ℛ with an involution and char ℛ ≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over ℛ are obtained. The previous known results on some systems of matrix equations are special cases of the new results.

[1]  Mark Yasuda,et al.  A Spectral Characterization of Generalized Real Symmetric Centrosymmetric and Generalized Real Symmetric Skew-Centrosymmetric Matrices , 2002, SIAM J. Matrix Anal. Appl..

[2]  Aaron Melman.  Symmetric centrosymmetric matrix-vector multiplication , 2000 .

[3]  Alan L. Andrew.  Classroom Note: Centrosymmetric Matrices , 1998, SIAM Rev..

[4]  A. Cantoni,et al.  Eigenvalues and eigenvectors of symmetric centrosymmetric matrices , 1976 .

[5]  Salvatore D. Morgera,et al.  On the reducibility of centrosymmetric matrices — Applications in engineering problems , 1989 .

[6]  Irwin S. Pressman,et al.  MATRICES WITH MULTIPLE SYMMETRY PROPERTIES : APPLICATIONS OF CENTROHERMITIAN AND PERHERMITIAN MATRICES , 1998 .

[7]  Qing-Wen Wang,et al.  Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra , 2002 .

[8]  R. D. Hill,et al.  On perhermitian matrices , 1990 .

[9]  F. Don.  On the symmetric solutions of a linear matrix equation , 1987 .

[10]  Qingwen Wang,et al.  A System of Matrix Equations and a Linear Matrix Equation Over Arbitrary Regular Rings with Identity , 2003 .

[11]  Russell M. Reid,et al.  Classroom Note: Some Eigenvalue Properties of Persymmetric Matrices , 1997, SIAM Rev..

[12]  Sujit Kumar Mitra,et al.  Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations , 1976 .

[13]  Dai Hua.  On the symmetric solutions of linear matrix equations , 1990 .

[14]  K. Chu.  Symmetric solutions of linear matrix equations by matrix decompositions , 1989 .

[15]  R. D. Hill,et al.  On centrohermitian matrices , 1990 .

[16]  A. C. Aitken.  Determinants and matrices , 1940 .

[17]  W. Vetter.  Vector structures and solutions of linear matrix equations , 1975 .

[18]  Anna Lee,et al.  Centrohermitian and skew-centrohermitian matrices , 1980 .

[19]  Jan R. Magnus,et al.  The Elimination Matrix: Some Lemmas and Applications , 1980, SIAM J. Algebraic Discret. Methods.

[20]  J. Weaver.  Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors , 1985 .