The bisymmetric solutions of the matrix equation [formula omitted] and its optimal approximation

Abstract

A matrix A = ( a ij ) ∈ R n × n is said to be bisymmetric matrix if a ij = a ji = a n + 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 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 bisymmetric matrix group [ X 1 ( 0 ) , X 2 ( 0 ) , … , X l ( 0 ) ] , 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 [ X ¯ 1 , X ¯ 2 , … , X ¯ l ] in Frobenius norm can be obtained by finding the least norm bisymmetric solution group of new matrix equation A 1 X ∼ 1 B 1 + A 2 X ∼ 2 B 2 + ⋯ + A l X ∼ l B l = C ∼ , where C ∼ = C - A 1 X ¯ 1 B 1 - A 2 X ¯ 2 B 2 - ⋯ - A l X ¯ l B l .