A generalized Hermite-Biehler theorem

TL;DR

This paper extends the Hermite-Biehler theorem to cases with a single interlacing break, enabling solutions to spectral problems involving specific matrix perturbations.

Contribution

It provides a full characterization of zero sets when interlacing is broken once, and applies this to solve spectral problems for certain matrix perturbations.

Findings

Characterization of zero sets with one interlacing break

Solution to spectral problems for rank-one multiplicative perturbations

Analysis of rank two additive perturbations of Jacobi matrices

Abstract

The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing is broken at exactly one location. Using this we solve the direct and inverse spectral problem for rank-one multiplicative perturbations of finite Hermitian matrices. We also treat certain rank two additive perturbations of finite Jacobi matrices.