An Iterative Method for the Generalized Bisymmetric Solution of Matrix Equation AXB=C

• 1、LanZhou university

Abstract：For fixed generalized reflection matrix P, i.e., P^T=P, P^2=I, then matrix X is said to be generalized bisymmetric, if $X=PXP$ and $X=X^T$. In this paper, an iterative method is established to solve the linear matrix equation $AXB=C$ over generalized bisymmetric $X$. For any initial generalized bisymmetric matrix $X_1$, when $AXB=C$ is consistent, we can obtain the generalized bisymmetric solution of the matrix equation AXB=C within finite iterative steps by the iteration method in the absence of roundoff errors; Moreover, the least-norm solution $X^*$ can be obtained by choosing a special kind of initial generalized bisymmetric matrix. In addition, the unique optimal approximation solution $\\\\\\\\\\\\\\\\hat X$ to given matrix $X_0 $ in Frobenius norm can be derived by finding the least-norm generalized bisymmetric solution $\\\\\\\\\\\\\\\\widetilde X^\\\\\\\\\\\\\\\\ast$ of the new matrix equation $A\\\\\\\\\\\\\\\\widetilde X B=\\\\\\\\\\\\\\\\widetilde C$, here, $\\\\\\\\\\\\\\\\widetilde X=X-X_0$, and $\\\\\\\\\\\\\\\\widetilde C=C-AX_0B$. Given numerical examples show that the algorithm is quite efficient.

keywords： Iterative method Generalized bisymmetric solution Least-norm solution Optimal approximation

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