The minimal critical sets of refined inertias for [formula omitted] full sign patterns

Abstract

A sign pattern (matrix) A is a matrix whose entries are from the set {+,−,0}. If no entry of A is zero, then A is called a full sign pattern. The inertia of a real matrix A is the ordered triple (n+,n−,n0), in which n+, n− and n0 are the numbers of its eigenvalues (counting multiplicities) with positive, negative and zero real parts, respectively. The refined inertia of A is the ordered 4-tuple (n+,n−,nz,2np) where nz (resp., 2np) is the number of zero (resp., nonzero pure imaginary) eigenvalues of A. An n×n sign pattern A is an inertially arbitrary pattern (IAP) if it allows all possible inertias. Similarly, A is a refined inertially arbitrary pattern (rIAP) if it allows all possible refined inertias. A proper subset S of the set of all possible inertias (resp., refined inertia) of real matrices of order n is called a critical set of inertias (resp., critical set of refined inertias) for a family F of sign patterns of order n if for every A∈F, S⊆i(A) ensures that A is inertially arbitrary (resp., S⊆ri(A) ensures that A is refined inertially arbitrary); S is called a minimal critical set of inertias (resp., minimal critical set of refined inertias) for F if no proper subset of S is a critical set of inertias (resp., refined inertias) for F. In this paper, all minimal critical sets of inertias and refined inertias for 3×3 full sign patterns are identified.