Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint

An n x n real matrix A = (a(i j))(n x n) is called bi-symmetric matrix if A is both symmetric and per-symmetric, that is, a(i j) = a (j i) and a(i j) = a(n+1) (- j.n+1-i) (i, j = 1, 2, . . . , n). This paper is mainly concerned with finding the least-squares bi-symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright (c) 2007 John Wiley & Sons, Ltd.