Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2

Ber-Lin Yu

doi:10.4236/alamt.2013.32002

Abstract

Let S be a nonempty, proper subset of all refined inertias. Then, S is called a critical set of refined inertias for ireducible sign patterns of order n if is sufficient for any sign pattern A to be refined inertially arbitrary. If no proper subset of Sis a critical set of refined inertias, then S is a minimal critical set of refined inertias for sign patterns of order n . In this paper, all minimal critical sets of refined inertias for irreducible sign patterns of order 2 are identified. As a by-product, a new approach is presented to identify all minimal critical sets of inertias for irreducible sign patterns of order 2.

Highlights

A n tries n sign from the pattern is a set, 0 matrix whereAdenotes a positive real number; see, e.g., [1]
If no proper subset of S is a critical set of refined inertias, S is a minimal critical set of refined inertias for sign patterns of order n
If no proper subset of S is a critical set of refined inertias, S is a minimal critical set of refined inertias for irreducible sign patterns of order n

Summary

Introduction

Denotes a positive (respectively, negative) real number; see, e.g., [1]. The set of all real matrices with the same sign pattern as the n n sign pattern Ais the qualitative class. We note that all minimal critical sets of inertias for irreducible sign patterns of order 2 have been identified in [6]. Identifying all minimal critical sets of inertias for irreducible zero-nonzero (or sign) patterns of order 3 has been posed as an open question in [6]. We concentrate on the minimal critical sets of refined inertias for irreducible sign patterns of order 2.

Preliminaries

Minimal Critical Sets of Inertias for Irreducible Sign Patterns of Order 2

Concluding Remarks