$\sigma$-Stable Matrices

TL;DR

This paper introduces $\sigma$-Stable matrices, linking polynomial roots to stability conditions and exploring implications for eigenvalues and coefficient behavior in matrix analysis.

Contribution

It presents the concept of $\sigma$-Stable matrices, proves related stability criteria, and discusses implications for eigenvalue analysis and coefficient scaling.

Findings

Real roots of characteristic polynomial coefficients indicate sign changes.

Largest real root corresponds to stability point when maximal eigenvalue is real.

Implications for coefficient behavior in matrix stability analysis.

Abstract

$\sigma$-Stable matrices are introduced and it is shown that the real roots of the polynomials comprising the coefficients of the characteristic polynomial indicate the coefficient sign changes. A proof of Obrechkoff is then used to show that the largest real root from the coefficients is the point of stability when the maximal eigenvalue of the $\sigma$-stable matrix is in $\mathbb{R}$. Some implications of the coefficient behaviour for a scaling relation are then discussed.