An iterative method for the least squares symmetric solution of the linear matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: min parallel to AXB - C parallel to with unknown symmetric matrix X. By this iterative method, for any initial symmetric matrix X-0, a Solution X* can be obtained within finite iteration steps in the absence of roundoff errors, and the solution X* with least norm call be obtained by choosing a special kind of initial symmetric matrix. In addition, the unique optimal approximation solution (X) over cap to a given matrix (X) over bar in Frobenius norm can be obtained by first finding the least norm solution T of the new minimum residual problem: min parallel to A (X) over tildeB - (C) over tilde parallel to with unknown symmetric matrix (X) over tilde, where (C) over tilde = C - A (X) over bar+(X) over bar (-T)/2 B. Given numerical examples are show that the iterative method is quite efficient. (c) 2005 Elsevier Inc. All rights reserved.