Refined inertias of positive and hollow positive patterns

Abstract

We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia (n+,n−,nz,2np)\left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n+≥1{n}_{+}\ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n+≥1{n}_{+}\ge 1 and n−≥2{n}_{-}\ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order nn, we show that as n→∞n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for nn odd and 3/4 for nn even.

Keywords

• refined inertia
• positive pattern
• hollow positive pattern
• circulant matrix
• 15a18
• 15b35
• 15b48