The Nearest Bisymmetric Solutions of Linear Matrix Equations (Journal of Computational Mathematics, 2021)
The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (I) $A_1X_1B_1+A_2X_2B_2+…+A_kX_kB_k=D$, (II) $A_1XB_1+A_2XB_2+…+A_kXB_k=D$ and (III) $(A_1XB_1, A_2XB_2, ··· , A_kXB_k) =(D_1, D_2, ··· , D_k)$ are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the nearest solutions are also provided.
Keywords: Bisymmetric matrix; Matrix equation; Matrix nearness problem; Kronecker product; Frobenius norm; Moore-Penrose generalized inverse.