Abstract
A matrix A ∈ R n × n is called a bisymmetric matrix if its elements a i , j satisfy the properties a i , j = a j , i and a i , j = a n - j + 1 , n - i + 1 for 1 ⩽ i , j ⩽ n . This paper considers least squares solutions to the matrix equation AX = B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.
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