**Title: Sign pattern matrices that require all distinct eigenvalues **

Reporter: Prof. Zhongshan Li ( Georgia State University )

Time: 2019-6-14 (Friday) 10:00

Place: G508

Abstract: A sign pattern (matrix) is a matrix whose entries are from the set {+, -, 0}. We say that a sign pattern A requires a certain matrix property P if every real matrix whose entries have signs agreeing with A has the property P. Some necessary or sufficient conditions for a square sign pattern to require all distinct eigenvalues are presented. Characterization of such sign pattern matrices is equivalent to determining when a certain real polynomial takes on only positive values whenever all of its variables are assigned arbitrarily chosen positive values. It is known that such sign patterns require a fixed number of real eigenvalues. The 3*3 irreducible sign patterns that require 3 distinct eigenvalues have been identified previously. We characterize the 4 *4 irreducible sign patterns that require four distinct real eigenvalues and those that require four distinct nonreal eigenvalues. The 4*4 irreducible sign patterns that require two distinct real eigenvalues and two distinct nonreal eigenvalues are investigated. Some related open problems are discussed.