A Note on the Minimum Cardinality of Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 4

If there exists a nonempty, proper subset S of the set of all (n+ 1)(n+ 2)/2 inertias such that S ⊆ i(A) is sufficient for any n×n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In [Kim, Olesky and Driessche, Critical sets of inertias for matrix patterns, Linear and Multilinear Algebra, 57 (3) (2009) 293-306], identifying all minimal critical sets of inertias for n×n zero-nonzero patterns with n ≥ 3 and the minimum cardinality of such a set are posed as two open questions by Kim, Olesky and Driessche. In this note, the minimum cardinality of all critical sets of inertias for 4× 4 irreducible zero-nonzero patterns is identified. Keywords—Zero-nonzero pattern, Inertia, Critical set of inertias, Inertially arbitrary.