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

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 SA. Then the best approximation problem for the inverse eigenproblem is discussed. That is: given an arbitrary A∗, find a matrix Â∈SA 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.