Critical sets of inertias for matrix patterns

Abstract

An n by n zero-nonzero (resp. sign) pattern is a matrix with entries in {*, 0} (resp. {+, −, 0}), where * denotes a nonzero real number. If allows all (n + 1)(n + 2)/2 possible inertias, then is inertially arbitrary. A nonempty, proper subset S of the set of all possible inertias is a critical set of inertias of order n provided an n by n pattern is inertially arbitrary whenever allows all inertias in S, and S is a minimal critical set if no proper subset of S is a critical set. This new concept of critical sets of inertias facilitates the identification of inertially arbitrary patterns. Minimal critical sets of inertias for irreducible zero-nonzero patterns of orders n = 2, 3, 4 are identified. Using these critical sets of inertias, it is shown that a direct sum of two non-inertially-arbitrary irreducible zero-nonzero patterns of orders ≤ 4 is not inertially arbitrary. However, it is shown that there exists a reducible inertially arbitrary zero-nonzero pattern that is a direct sum of non-inertially-arbitrary irreducible zero-nonzero patterns of order 5. Minimal critical sets of inertias for irreducible sign patterns of orders n = 2, 3 are also identified. It is shown that in general critical sets of inertias for irreducible sign patterns are different from those of irreducible zero-nonzero patterns.