Inverse spectral problem for normal matrices and a generalization of the Gauss-Lucas theorem

TL;DR

This paper extends classical theorems related to the roots of polynomials and eigenvalues of normal matrices, providing new insights into inverse spectral problems and generalizing the Gauss-Lucas theorem.

Contribution

It introduces an analog of the Cauchy-Poincare theorem for normal matrices, solves an inverse spectral problem, and generalizes the Gauss-Lucas theorem with applications to longstanding conjectures.

Findings

Established a majorization-based separation theorem for normal matrices

Provided a solution to the inverse spectral problem for normal matrices

Generalized the Gauss-Lucas theorem and proved related conjectures

Abstract

We estabish an analog of the Cauchy-Poincare separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borg-type result) for a normal matrix. Using this result we essentially generalize and complement the known Gauss--Lucas theorem on the geometry of the roots of a complex polynomial and of its derivative. In turn the last result is applied to prove the old conjectures of de Bruijn-Springer and Schoenberg about these roots.