Positive-Definite Matrices over Finite Fields

TL;DR

This paper explores the concept of positive-definite matrices within finite fields, examining how classical properties and equivalences of such matrices extend or differ in this algebraic setting.

Contribution

It introduces the notion of positive-definiteness for matrices over finite fields and analyzes which classical properties are preserved or altered.

Findings

Classical equivalences partially extend to finite fields

Certain properties of positive-definite matrices do not hold in finite fields

Framework for defining positive-definiteness over finite fields established

Abstract

The study of positive-definite matrices has focused on Hermitian matrices, that is, square matrices with complex (or real) entries that are equal to their own conjugate transposes. In the classical setting, positive-definite matrices enjoy a multitude of equivalent definitions and properties. In this paper, we investigate when a square, symmetric matrix with entries coming from a finite field can be called "positive-definite" and discuss which of the classical equivalences and implications carry over.