The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation

Abstract

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenproblem defined as follows: given a set of complex n-vectors {x i} i=1 m and a set of complex numbers {λ i} i=1 m and an s-by-s real matrix Co, find an n-by-n real centrosymmetric matrix C such that the s-by-s leading principal submatrix of C is C 0, and {x i} i=1 m and {λ i} i=1 m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix C̃, find a matrix C which is the solution to the constrained inverse problem such that the distance between C and C̃ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. Some illustrative experiments are also presented. © 2005 Society for Industrial and Applied Mathematics.