#### Date of Award

Summer 8-11-2011

#### Degree Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematics and Statistics

#### First Advisor

Prof. Frank Hall

#### Second Advisor

Prof. Zhongshan Li

#### Third Advisor

Prof. Marina Arav

#### Abstract

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive(respectively, negative, zero) entry of B by + (respectively, -, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B: sgn(B) = A}.

An n by n sign pattern matrix A requires all distinct eigenvalues if every real matrix whose sign pattern is represented by A has n distinct eigenvalues. In this thesis, a number of sufficient and/or necessary conditions for a sign pattern to reuiqre all distinct eigenvalues are reviewed. In addition, for n=2 and 3, the n by n sign patterns that require all distinct eigenvalues are surveyed. We determine most of the 4 by 4 irreducible sign patterns that require four distinct eigenvalues.

#### Recommended Citation

Kim, Paul J., "On the 4 by 4 Irreducible Sign Pattern Matrices that Require Four Distinct Eigenvalues." Thesis, Georgia State University, 2011.

https://scholarworks.gsu.edu/math_theses/101