Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A(1)X(1)B(1) + A(2)X(2)B(2) = C
A matrix P is an element of R-nxn is called a generalized reflection matrix if P-T = P and P-2 = I. An n x n matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP. (A = -PAP). In this paper, three iterative algorithms are proposed for solving the linear matrix equation A(1)X(1)B(1) + A(2)X(2)B(2) = C over reflexive (anti-reflexive) matrices X-1 and X-2. When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive ( anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper. (C) 2009 Elsevier Ltd. All rights reserved.
iterative algorithm matrix equation generalized reflection matrix anti-reflexive matrix reflexive matrix