An inverse eigenproblem and an associated approximation problem for generalized reflexive and anti-reflexive matrices

In this paper, we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows: given a set of real n -vectors { x i } i = 1 m and a set of real numbers { λ i } i = 1 m , and an n -by- n real generalized reflexive matrix A (or generalized anti-reflexive matrix B ) such that { x i } i = 1 m and { λ i } i = 1 m are the eigenvectors and eigenvalues of A (or B ), respectively, we solve the best approximation problem for the inverse eigenproblem. That is, given an arbitrary real n -by- n matrix A ̃ , we find a matrix A A ̃ which is the solution to the inverse eigenproblem such that the distance between A ̃ and A A ̃ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive (or generalized anti-reflexive) matrices. Two numerical examples are also presented to show that our method is effective.