Some maximal absolutely unstable sign patterns of 4×4 matrices

Abstract

In this paper we discuss the absolute instability of the 4×4 qualitative matrices whose diagonal elements are 0, 0, 0 and —. A matrix is said to be stable if all its characteristic values have negative real parts. A sign pattern ( +, −, 0) of a matrix is said to be absolutely unstable if there is no stable matrix which has the sign pattern. Of 531441 different patterns, 299953 are found to be absolutely unstable. By using the overpattern-sub-pattern concept we can define an order relation in the set of qualitative matrices and it is found that there arc 12 basic absolutely unstable sign patterns which are maximal.