Least–Squares Solution of Inverse Problem for Hermitian Anti–reflexive Matrices and Its Appoximation

In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti–reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X,B we have minA ‖AX −B‖. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A*, find a matrix  ∈ SE which is nearest to A* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.