Refined inertias of positive and hollow positive patterns

Abstract

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 − , n z , 2 n p ) \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 n n , we show that as n → ∞ n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.