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The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CY T D = E
Author(s) -
Hajarian Masoud,
Dehghan Mehdi
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1459
Subject(s) - mathematics , symmetric matrix , matrix norm , matrix (chemical analysis) , centrosymmetric matrix , elementary symmetric polynomial , norm (philosophy) , nonnegative matrix , mathematical analysis , combinatorics , eigenvalues and eigenvectors , orthogonal polynomials , materials science , physics , quantum mechanics , political science , law , discrete orthogonal polynomials , composite material
An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P −1 = P T . 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 equationand the minimum Frobenius norm residual problemover 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 Y 0 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.