Sign structures of 3 × 3 stable matrices and their generalization to higher‐order matrices

Abstract

AbstractA matrix A is called stable if all its characteristic roots have negative real parts. This paper discusses the stability of sign matrices (matrices for which only the sign of the elements is given) and presents some basic sign patterns of stable matrices. The stability of a matrix and the stability of a system whose interaction matrix coincides with the matrix are related closely and thus the sign pattern consideration is a useful method to study the internal structures of stable systems. We discuss the potential stability (the property that there exists a stable matrix with the sign pattern) and the sign instability (the property that there are no stable matrices with the sign pattern) of sign matrices. A signed digraph is defined for each sign matrix and the connection between its structure and the matrix stability is discussed. Methods to check potential stability and sign instability of sign matrices are given. Through the computer stability check of all 3x3 matrices, the relationships between graph structure and matrix stability are clarified and seven basic sign structures of 3x3 stable matrices are obtained. Finally, based upon these seven basic structures, potentially stable structures of higher‐order matrices are considered. Some stable sign patterns of nxn matrices together with their stability proofs are given.