The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P−1 = PT. An n × n real matrix Y is called a generalized centro-symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro-symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro-symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro-symmetric matrix, a generalized centro-symmetric solution (least squares generalized centro-symmetric solution) can be obtained within a finite number of iterations in the absence of round-off errors, and the least Frobenius norm generalized centro-symmetric solution (the minimal Frobenius norm least squares generalized centro-symmetric solution) can be derived by choosing a special kind of initial generalized centro-symmetric matrices. We also obtain the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix Y0 in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.