The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation

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 {xi}i=1 and a set of complex numbers {λi}i=1, and an s-by-s real matrix C0, find an n-by-n real centrosymmetric matrix C such that the s-by-s leading principal submatrix of C is C0, and {xi}i=1 and {λi}i=1 are the eigenvectors and eigenvalues of C respectively. We 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.