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

Abstract: 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.

• The paper shows that classical definitions of positive definiteness and semidefiniteness become distinct matrix classes over finite fields, then classifies their inclusions and derives exact counts, bounds, or generating functions for each type.
• The authors use finite-field counting, congruence classifications, orthogonal-group actions, and Dwork’s rationality theorem to analyze difficult cases, including positive semidefinite and totally positive matrices.
• The results establish formulas such as the type-1 positive-definite count and rank-based semidefinite counts, while identifying open problems involving quasipolynomial behavior, minimal field sizes, and unresolved inclusion relations.

The paper develops a systematic theory of positive definite, positive semidefinite, and totally positive matrices over finite fields $\mathbb{F}_q$, building on the definition of a positive element of $\mathbb{F}_q^\times$ as a nonzero square due to Cooper–Hanna–Whitlatch (2608.17702). Over $\mathbb{R}$ and $\mathbb{C}$, the classical characterizations of positive definiteness — positivity of leading principal minors, Cholesky factorization $A = LL^*$, the Gram matrix property $A = B^*B$, and unitary diagonalization with positive spectrum — are all equivalent. The central observation of this work is that these equivalences collapse over finite fields: each criterion yields a distinct class of matrices, which the authors label types 1 through 4 for positive definiteness and types 1 through 5 for positive semidefiniteness. For each type they derive explicit enumeration formulas or bounds, relate the classes via Venn diagrams, and apply Dwork's rationality theorem to obtain structural formulas in the cases resisting closed-form treatment.

Positivity over finite fields

An element $a \in \mathbb{F}_q^\times$ is positive if it is a square; in characteristic two every nonzero element is positive, while in odd characteristic exactly half are. A field is definite if every positive element has a positive square root, which holds precisely when $q \equiv 3 \pmod{4}$ or $q$ is even. Two elementary counting facts drive much of the paper: by a pigeonhole argument, every element of an odd-order field is a sum of two squares, and every positive element is a sum of two negative elements. The authors also prove that the number of $y \in \mathbb{F}_q^+$ with $\mathbb{F}_q^\times$0 equals $\mathbb{F}_q^\times$1 when $\mathbb{F}_q^\times$2 and $\mathbb{F}_q^\times$3 when $\mathbb{F}_q^\times$4; this count also determines the number of common neighbors of adjacent vertices in Paley graphs.

Positive definite matrices

Four inequivalent definitions are studied:

• Type 1: all leading principal minors are positive.
• Type 2: $\mathbb{F}_q^\times$5 with lower triangular $\mathbb{F}_q^\times$6 having positive diagonal entries (a finite-field Cholesky condition).
• Type 3: $\mathbb{F}_q^\times$7 for some invertible $\mathbb{F}_q^\times$8 (Gram matrix).
• Type 4: orthogonally similar to a diagonal matrix with positive diagonal entries.

For type 1, the count is $\mathbb{F}_q^\times$9 for odd $\mathbb{R}$0 and $\mathbb{R}$1 for even $\mathbb{R}$2 (2608.17702). For type 2, the classes coincide on definite fields; over indefinite fields the authors prove a new formula, $\mathbb{R}$3, obtained by showing that two lower triangular matrices with positive diagonal yield the same product $\mathbb{R}$4 exactly when they differ by right multiplication by a sign diagonal matrix — equivalently, by a coset count via Lagrange's theorem. Type 3 counts go back to MacWilliams and Dickson, using the classification of symmetric matrices under congruence. Type 4 is new: generalizing a lemma of Kung from $\mathbb{R}$5 to arbitrary subgroups, the authors express $\mathbb{R}$6 as a sum over compositions of $\mathbb{R}$7 of orbit sizes under orthogonal conjugation, yielding the generating-function identity

$\mathbb{R}$8

where $\mathbb{R}$9. The inclusion structure among the four types is worked out explicitly, with constructive witnesses for every region of the corresponding Venn diagrams. Notably, the authors point out an error in Cooper–Hanna–Whitlatch's claim that types 1, 2, and 3 coincide on definite fields: a Gram matrix built from linearly independent vectors need not have positive leading principal minors.

Positive semidefinite matrices

Five types arise here. Type 1 is defined via principal submatrices: for each $\mathbb{C}$0 there exists $\mathbb{C}$1 such that $\mathbb{C}$2 is $\mathbb{C}$3 and all larger principal submatrices are singular. The authors first establish, as an independent contribution valid over $\mathbb{C}$4, a new characterization of psd matrices (condition (f) in their Theorem 2.2) via Bocher's theorem and Cauchy interlacing. They then derive a recurrence for $\mathbb{C}$5 whose solution is $\mathbb{C}$6 for odd $\mathbb{C}$7. Type 2 (Cholesky with nonnegative diagonal) admits only upper bounds, e.g. $\mathbb{C}$8 for $\mathbb{C}$9, derived from the $A = LL^*$0-binomial theorem together with sign-change identifications. On definite fields the inclusion reverses relative to the definite case: $A = LL^*$1.

For type 3, the authors complete MacWilliams' program by proving that when $A = LL^*$2 is odd and $A = LL^*$3, every symmetric matrix of rank $A = LL^*$4 is psd of type 3 — this uses the fact that any negative element is a sum of two squares, so the "bad" congruence class $A = LL^*$5 with $A = LL^*$6 negative still factors as $A = LL^*$7. Thus $A = LL^*$8 for $A = LL^*$9, where $A = B^*B$0 counts symmetric matrices of rank $A = B^*B$1. Type 4 counts mirror the type-4 positive definite case with one extra zero eigenvalue allowed, giving a generating function raised to the power $A = B^*B$2.

Type 5 (all principal minors nonnegative) is trivial in characteristic two — every symmetric matrix qualifies — but subtle for odd $A = B^*B$3. The authors compute $A = B^*B$4 exactly, distinguishing $A = B^*B$5 and $A = B^*B$6, and prove that for $A = B^*B$7, $A = B^*B$8, so all five notions nest appropriately in that case; for $A = B^*B$9 there is no inclusion between $a \in \mathbb{F}_q^\times$0 and types 1 or 2. For $a \in \mathbb{F}_q^\times$1 no closed form is known; instead, the set $a \in \mathbb{F}_q^\times$2 is decomposed into $a \in \mathbb{F}_q^\times$3 pieces according to which principal minors vanish versus are positive, each piece realized as a projection of an algebraic set (with auxiliary variables $a \in \mathbb{F}_q^\times$4 satisfying $a \in \mathbb{F}_q^\times$5, fibers of size $a \in \mathbb{F}_q^\times$6). Applying Dwork's rationality theorem yields a structural formula: the count over $a \in \mathbb{F}_q^\times$7 is $a \in \mathbb{F}_q^\times$8 for algebraic numbers $a \in \mathbb{F}_q^\times$9. Based on this and computations, the authors conjecture a quasipolynomial behavior: degree-$q \equiv 3 \pmod{4}$0 rational polynomials $q \equiv 3 \pmod{4}$1 with the count equal to $q \equiv 3 \pmod{4}$2 when $q \equiv 3 \pmod{4}$3.

Totally positive matrices

A matrix is totally positive if every minor is positive. Scaling rows and columns by positive diagonals preserves total positivity, and each equivalence class contains a unique elementary totally positive matrix (first row and column all ones), giving $q \equiv 3 \pmod{4}$4. Exact formulas follow for small orders: for instance, $q \equiv 3 \pmod{4}$5 for even $q \equiv 3 \pmod{4}$6 and $q \equiv 3 \pmod{4}$7 in characteristic two. For odd $q \equiv 3 \pmod{4}$8, SageMath computations show $q \equiv 3 \pmod{4}$9 for $q$0, with the first nonzero values at $q$1.

Two structural results constrain existence. First, when $q$2, any nontrivial row or column permutation destroys total positivity, since the relevant $q$3 minors become negative. Second, a monotonicity argument on ordered subsets of positive elements yields necessary conditions: if an $q$4 totally positive matrix exists with $q$5, then $q$6 for even $q$7, $q$8 for $q$9, and $y \in \mathbb{F}_q^+$0 for $y \in \mathbb{F}_q^+$1. The authors note plainly that their proof uses only order-2 minors of a restricted form, so these bounds are likely far from optimal. As with $y \in \mathbb{F}_q^+$2, Dwork's theorem gives structural formulas for the count over extension fields $y \in \mathbb{F}_q^+$3 — of the form $y \in \mathbb{F}_q^+$4 for even $y \in \mathbb{F}_q^+$5, and $y \in \mathbb{F}_q^+$6 times such a sum for odd $y \in \mathbb{F}_q^+$7 — supporting a conjectured quasipolynomial formula with polynomials of degree $y \in \mathbb{F}_q^+$8.

Limitations and open questions

Several gaps are acknowledged explicitly. No closed forms exist for $y \in \mathbb{F}_q^+$9 (only bounds), for $\mathbb{F}_q^\times$00 with $\mathbb{F}_q^\times$01 over odd $\mathbb{F}_q^\times$02, or for $\mathbb{F}_q^\times$03 beyond small orders; the quasipolynomial conjectures remain unproven. The Venn diagrams for the psd types are presented as heuristic rather than fully verified — for example, it is unknown whether matrices exist that are both $\mathbb{F}_q^\times$04 and $\mathbb{F}_q^\times$05 but neither $\mathbb{F}_q^\times$06 nor $\mathbb{F}_q^\times$07. The growth rate of the minimal field size admitting an $\mathbb{F}_q^\times$08 totally positive matrix is posed as an open question, with the current bounds expected to be improvable.

Conclusion

This paper demonstrates that the classical equivalences among definitions of matrix positivity fragment over finite fields into genuinely distinct combinatorial objects, and it supplies exact enumerations for most of them. The combination of elementary finite-field counting, group actions of orthogonal groups, and arithmetic-geometric input via Dwork's theorem provides both concrete formulas and a structural framework, leaving the quasipolynomial conjectures and the minimal-field-size question as well-defined targets for further work.