The bisymmetric solutions of the matrix equation A1X1B1+A2X2B2+⋯+AlXlBl=C and its optimal approximation

A matrix A=(aij)∈Rn×n is said to be bisymmetric matrix if aij=aji=an+1-j,n+1-i for all 1⩽i,j⩽n. In this paper, an iterative method is constructed to find the bisymmetric solutions of matrix equation A1X1B1+A2X2B2+⋯+AlXlBl=C where [X1,X2,…,Xl] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial bisymmetric matrix group , a bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm bisymmetric solution group can be obtained by choosing a special kind of initial bisymmetric matrix group. In addition, the optimal approximation bisymmetric solution group to a given bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm bisymmetric solution group of new matrix equation , where .