Total nonnegativity of finite Hurwitz matrices and root location of polynomials

TL;DR

This paper investigates the relationship between the total nonnegativity of finite Hurwitz matrices and the root locations of real polynomials, providing necessary and sufficient conditions for total nonnegativity.

Contribution

It offers a detailed explanation of the phenomenon and establishes exact criteria linking polynomial roots in the left half-plane to total nonnegativity of Hurwitz matrices.

Findings

Necessary and sufficient conditions for total nonnegativity.

Clarification of the non-equivalence between quasi-stability and Hurwitz matrix properties.

Insight into root location implications for matrix positivity.

Abstract

In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the \emph{closed} left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.