Positive-Definite Matrices over Finite Fields

2022/02/08 by Cooper, Joshua, Hanna, Erin, Whitlatch, Hays · 4 citations
#05C50 #15A23 #15B33 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2202.04012

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.

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• Counting Cholesky factorizations of the zero matrix over \mathbbF₂
• The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings
• Cholesky decomposition for symmetric matrices over finite fields
• Entrywise transforms preserving matrix positivity and non-positivity

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