The Nearest Bisymmetric Solutions of Linear Matrix Equations

Authors

  • Zhenyun Peng
  • Xiyan Hu
  • Lei Zhang

Keywords:

Bisymmetric matrix, Matrix equation, Matrix nearness problem, Kronecker product, Frobenius norm, Moore-Penrose generalized inverse.

Abstract

The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (I) $A_1X_1B_1+A_2X_2B_2+\cdots+A_kX_kB_k=D$, (II) $A_1XB_1+A_2XB_2+\cdots+A_kXB_k=D$ 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.

Published

2021-07-01

Issue

Section

Articles