Linear Algebra Seminar

Speaker: Wei Gao

Title: Tree Sign Patterns that Require H_n

Abstract: A sign pattern (matrix) is a matrix whose entries are from the set \({+, -, 0}\). The refined inertia of a square real matrix is the ordered 4-tuple \((n_+, n_-, n_z, 2n_p)\), where \(n_+ (resp., n_-)\) is the number of eigenvalues with positive (resp., negative) real part, \(n_z\) is the number of zero eigenvalues and \(2n_p\) is the number of pure imaginary eigenvalues.

The set of refined inertias \(H_n=(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\) is important for the onset of Hopf bifurcation in dynamical systems.

Bodine et al.  conjectured that no irreducible sign pattern that requires \(H_n\) exists for n sufficiently large. In this talk, we discuss the star and path sign patterns that require \(H_n\). It is shown that for each \(n\ge 5\), a star sign pattern requires \(H_n\) if and only if it is equivalent to one of the five sign patterns identified in the talk. This resolves the above conjecture. It is also shown that no path sign pattern of order \(n\ge 5\) requires \(H_n\).