On the Stability of P-Matrices

Abstract

We establish two sufficient conditions for the stability of a $P$-matrix. First, we show that a $P$-matrix is positive stable if its skew-symmetric component is sufficiently smaller (in matrix norm) than its symmetric component. This result generalizes the fact that symmetric $P$-matrices are positive stable, and is analogous to a result by Carlson which shows that sign symmetric $P$-matrices are positive stable. Second, we show that a $P$-matrix is positive stable if it is strictly row (column) square diagonally dominant for every order of minors. This result generalizes the fact that strictly row diagonally dominant$P$-matrices are stable. We compare our sufficient conditions with the sign symmetric condition and demonstrate that these conditions do not imply each other.