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

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: min∥AXB − C∥ with unknown symmetric matrix X. By this iterative method, for any initial symmetric matrix X0, a solution X* can be obtained within finite iteration steps in the absence of roundoff errors, and the solution X* with least norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, the unique optimal approximation solution X^ to a given matrix X¯ in Frobenius norm can be obtained by first finding the least norm solution X∼∗ of the new minimum residual problem: min‖AX∼B-C∼‖ with unknown symmetric matrix X∼, where C∼=C-AX¯+X¯T2B. Given numerical examples are show that the iterative method is quite efficient.