Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F-norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above-mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  Kenneth L. Bowers,et al.  Sinc methods for quadrature and differential equations , 1987 .

[2]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[3]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

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

[5]  Michael K. Ng,et al.  Preconditioners for nonsymmetric block toeplitz-like-plus-diagonal linear systems , 2003, Numerische Mathematik.

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

[7]  Lei Zhang,et al.  Least Squares Solution of BXAT=T over Symmetric, Skew-Symmetric, and Positive Semidefinite X , 2003, SIAM J. Matrix Anal. Appl..

[8]  Sujit Kumar Mitra.  Common solutions to a pair of linear matrix equations A 1 XB 1 = C 1 and A 2 XB 2 = C 2 , 1973 .

[9]  Jan R. Magnus,et al.  L-structured matrices and linear matrix equations , 1983 .

[10]  O. Axelsson.  Iterative solution methods , 1995 .

[11]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[12]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[13]  A. Bjerhammar,et al.  Rectangular reciprocal matrices, with special reference to geodetic calculations , 1951 .

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

[15]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[16]  John E. Mottershead,et al.  Finite Element Model Updating in Structural Dynamics , 1995 .

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

[18]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[19]  Michael K. Ng,et al.  A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc-Galerkin linear systems , 2003 .

[20]  S. Mitra.  The matrix equations AX = C, XB = D , 1984 .

[21]  Shufang Xu,et al.  An Introduction to Inverse Algebraic Eigenvalue Problems , 1999 .

[22]  S. R. Searle,et al.  Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics , 1979 .

[23]  Sujit Kumar Mitra,et al.  A pair of simultaneous linear matrix equations A1XB1 = C1, A2XB2 = C2 and a matrix programming problem , 1990 .

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

[25]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[26]  Gene H. Golub,et al.  Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems , 2005, SIAM J. Sci. Comput..

[27]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[28]  C. R. Rao,et al.  Generalized Inverse of Matrices and its Applications , 1972 .

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