An efficient iterative method for solving the matrix equation AXB + CYD = E

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Abstract

Abstract This paper presents an iterative method for solving the matrix equation AXB + CYD = E with real matrices X and Y . By this iterative method, the solvability of the matrix equation can be determined automatically. And when the matrix equation is consistent, then, for any initial matrix pair [ X 0 , Y 0 ], a solution pair can be obtained within finite iteration steps in the absence of round‐off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair [ X̄ , Ȳ ] in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation AX̃B + CỸD = Ẽ , where Ẽ = E − AX̄B − CȲD . The given numerical examples show that the iterative method is efficient. Copyright © 2005 John Wiley & Sons, Ltd.

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