The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation
Zheng‐Jian Bai · SIAM Journal on Matrix Analysis and Applications · 2005
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 $\{\bx_i\}_{i=1}^{m}$ and a set of complex numbers $\{\la_i\}_{i=1}^m$, and an s-by-s real matrix $C_0$, find an n-by-n real centrosymmetric matrix C such that the s-by-s leading principal submatrix of C is $C_0$, and $\{\bx_i\}_{i=1}^{m}$ and $\{\la_i\}_{i=1}^m$ 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 $\tilde{C}$, find a matrix C which is the solution to the constrained inverse problem such that the distance between C and $\tilde{C}$ 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.