The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation

Abstract

In this paper, we first consider the inverse eigenvalue problem as follows: Find a matrix A with specified eigenpairs, where A is a Hermitian anti-reflexive matrix with respect to a given generalized reflection matrix J. The sufficient and necessary conditions are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by S A . Then the best approximation problem for the inverse eigenproblem is discussed. That is: given an arbitrary A ∗ , find a matrix A ̂ ∈S A which is nearest to A ∗ in the Frobenius norm. We show that the best approximation is unique and provide an expression for this nearest matrix.