The generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + ⋯ + A l X l B l = C and its optimal approximation

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 = X T = PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + ⋯ + A l X l B l = C where [X 1,X 2, ⋯ ,X l ] 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 generalized bisymmetric matrix group $\left[X_1^{(0)},X_2^{(0)},\cdots,X_l^{(0)}\right]$ , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized bisymmetric matrix group $\left[\overline{X}_1,\overline{X}_2,\cdots,\overline{X}_l\right]$ in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation $A_1\widetilde{X}_1B_1+A_2\widetilde{X}_2B_2+\cdots+A_l\widetilde{X}_lB_l=\widetilde{C}$ , where $\widetilde{C}=C-A_1\overline{X}_1B_1-A_2\overline{X}_2B_2-\cdots-A_l\overline{X}_lB_l$ . Given numerical examples show that the algorithm is efficient.