POSITIVE-DEFINITE MATRICES OVER FINITE FIELDS

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. 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.