# Global least squares solution of matrix equation $\sum_{j=1}^s A_jX_jB_j = E$

Document Type : Research Article

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Abstract

In this paper, an iterative method is proposed for solving matrix equation $\sum_{j=1}^s A_jX_jB_j = E$. This method is based on the global least squares (GL-LSQR) method for solving the linear system of equations with the multiple right hand sides. For applying the GL-LSQR algorithm to solve the above matrix equation, a new linear operator, its adjoint and a new inner product are de ned. It is proved that the new iterative method obtains the least norm solution of the mentioned matrix equation within finite iteration steps in the exact arithmetic, when the above matrix equation is consistent. Moreover, the optimal approximate solution $(X_1^* ,X_2^* ,\ldots,X_s^*)$ to a given multiple matrices $( \bar{X}_1, \bar{X}_2,\ldots,\bar{X}_s)$ can be derived by nding the least norm solution of a new matrix equation. Finally, some numerical experiments are given to illustrate the eciency of the new method.