A generalized inverse eigenvalue problem and $m$-functions

TL;DR

This paper studies a generalized inverse eigenvalue problem involving matrix pencils related to rational interpolation, providing new characterizations of eigenvectors and conditions for positive-definiteness, with a focus on $m$-functions for matrix reconstruction.

Contribution

It introduces a novel approach to reconstruct Hermitian matrices from eigenvalue problems using $m$-functions and characterizes associated rational functions and positivity conditions.

Findings

Reconstruction of Hermitian matrices from eigenvalues using matrix pencils.

Characterization of rational functions in eigenvector components.

Conditions for positive-definiteness of involved matrices.

Abstract

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ of matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix $\mathcal{H}_{[0,n]}$ with the entries $b_j's$, characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of $\mathcal{J}_{[0,n]}$ and which is often an assumption in the direct problem is also isolated. Further, the reconstruction of $\mathcal{H}_{[0,n]}$ is viewed through the inverse of the pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ which involves the concept of $m$-functions.