Positive definite, positive semidefinite and totally positive matrices over finite fields

Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper--Hanna--Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.