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 nn-vectors {xi}i=1m and a set of real numbers {λi}i=1m, and an nn-by-nn real generalized reflexive matrix AA (or generalized anti-reflexive matrix BB) such that {xi}i=1m and {λi}i=1m are the eigenvectors and eigenvalues of AA (or BB), respectively, we solve the best approximation problem for the inverse eigenproblem. That is, given an arbitrary real nn-by-nn matrix Ã, we find a matrix Aà which is the solution to the inverse eigenproblem such that the distance between à and Aà 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.