Date of Award

2018

Document Type

Open Access Thesis

Department

Mathematics

Sub-Department

College of Arts and Sciences

First Advisor

Joshua Cooper

Abstract

Positive definite matrices make up an interesting and extremely useful subset of Hermitian matrices. They are particularly useful in exploring convex functions and finding minima for functions in multiple variables. These matrices admit a plethora of equivalent statements and properties, one of which is an existence of a unique Cholesky decomposition. Positive definite matrices are not usually considered over finite fields as some of the definitions and equivalences are quickly seen to no longer hold. Motivated by a result from the theory of pressing sequences, which almost mirrors an equivalent statement for positive definite Hermitian matrices, we consider whether any of the theory of positive definiteness can be analogized for matrices over finite fields. New definitions are formed based on this motivation to be able to discuss positive definiteness in certain finite fields, relying heavily on the notion of the existence of a unique Cholesky decomposition. We explore what equivalences of positive definite Hermitian matrices can be analogized and present counterexamples for those which are still seen to fail. The final result not only holds for finite fields, but a certain subset of fields with a desired property.

Included in

Mathematics Commons

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