Article ID Journal Published Year Pages File Type
4598669 Linear Algebra and its Applications 2016 23 Pages PDF
Abstract

The refined inertia of a square real matrix B  , denoted ri(B)ri(B), is the ordered 4-tuple (n+(B),n−(B),nz(B),2np(B)), where n+(B)n+(B) (resp., n−(B)n−(B)) is the number of eigenvalues of B   with positive (resp., negative) real part, nz(B)nz(B) is the number of zero eigenvalues of B  , and 2np(B)2np(B) is the number of pure imaginary eigenvalues of B  . For n≥3n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)}Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. An n×nn×n sign pattern AA is said to require HnHn if Hn={ri(B)|B∈Q(A)}Hn={ri(B)|B∈Q(A)}. The star sign patterns of order n≥5n≥5 that require HnHn are characterized. More specifically, it is shown that for each n≥5n≥5, a star sign pattern requires HnHn if and only if it is equivalent to one of the five sign patterns identified in the paper.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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