# Irreducible sign patterns that require all distinct eigenvalues

## Description

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 arbitrary positive values. It is known that such sign patterns require a fixed number of real eigenvalues. The 3x3 irreducible sign patterns that require 3 distinct eigenvalues have been identified previously. We characterize the 4x4 irreducible sign patterns that require four distinct real eigenvalues and those that require four distinct nonreal eigenvalues. The 4x4 irreducible sign patterns that require two distinct real eigenvalues and two distinct nonreal eigenvalues are investigated. Joint work with: Yubin Gao, Victor Bailey, Frank Hall, Paul Kim.

## Date of Event

March 1, 2019

## Location

Parker Building, Room 301

Irreducible sign patterns that require all distinct eigenvalues

Parker Building, Room 301

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 arbitrary positive values. It is known that such sign patterns require a fixed number of real eigenvalues. The 3x3 irreducible sign patterns that require 3 distinct eigenvalues have been identified previously. We characterize the 4x4 irreducible sign patterns that require four distinct real eigenvalues and those that require four distinct nonreal eigenvalues. The 4x4 irreducible sign patterns that require two distinct real eigenvalues and two distinct nonreal eigenvalues are investigated. Joint work with: Yubin Gao, Victor Bailey, Frank Hall, Paul Kim.

## Presenter Bio

I grew up in Lanzhou, China. In 1983, I obtained my B.Sc. in mathematics from Lanzhou University. Then I spent three years at Beijing Normal University, graduating with a M.Sc. in Mathematics in 1986. After teaching briefly at Hebei Normal College, I went to North Carolina State University, where I got my Ph.D. in mathematics in 1990. For further personal information, you may read my mini autobiography