The solvability conditions for the inverse eigenvalue problem of Hermitian and generalized skew-Hamiltonian matrices and its approximation

摘要

In this paper, we first consider the inverse eigenvalue problem as follows: find a matrix A with specified eigenpairs, where A is a Hermitian and generalized skew-Hamiltonian matrix. 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 L_s. Then we discuss the best approximation problem for the inverse eigenproblem. that is, given an arbitrary A, find a matrix A~* ∈ L_s 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.