The minimal critical sets of refined inertias for 3×33×3 full sign patterns

A sign pattern (matrix) AA is a matrix whose entries are from the set {+,−,0}{+,−,0}. If no entry of AA is zero, then AA is called a full sign pattern. The inertia of a real matrix A   is the ordered triple (n+,n−,n0)(n+,n−,n0), in which n+n+, n−n− and n0n0 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)(n+,n−,nz,2np) where nznz (resp., 2np2np) is the number of zero (resp., nonzero pure imaginary) eigenvalues of A  . An n×nn×n sign pattern AA is an inertially arbitrary pattern (IAP) if it allows all possible inertias. Similarly, AA 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 FF of sign patterns of order n   if for every A∈FA∈F, S⊆i(A)S⊆i(A) ensures that AA is inertially arbitrary (resp., S⊆ri(A)S⊆ri(A) ensures that AA is refined inertially arbitrary); S   is called a minimal critical set of inertias (resp., minimal critical set of refined inertias) for FF if no proper subset of S   is a critical set of inertias (resp., refined inertias) for FF. In this paper, all minimal critical sets of inertias and refined inertias for 3×33×3 full sign patterns are identified.