$\sigma$-Stable Matrices

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.