Qualitative stability properties of matrices

Abstract

A matrix A is sign stable if some matrix with the same sign pattern of positive, negative, and zero entries has all eigenvalues with negative real parts. It is potentially stable if it is not sign unstable. The system A x + b = 0 is positively sign solvable if the sign pattern of the solution vector x contains all positive entries and is determined by the sign patterns of A and b.The characterization of potentially stable matrices remains unresolved. Counterexamples are presented refuting some previous results of Quirk and Campbell. New classes of potentially stable matrices are defined by means of recursive constructions, and a new necessary condition for potential stability is given.A characterization is presented of systems A x + b = 0 for which A is sign stable and the system is positively sign solvable. This leads to a recognition algorithm of time complexity which is linear in the order of A and the number of nonzero entries of A. The relation between this characterization and previous results of Manber is explored.