Positivity preservers over finite fields II

Abstract

We say that a matrix over a finite field ${\mathbb{F}}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in ${\mathbb{F}}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n({\mathbb{F}}_q)$ were classified for every $n\ge 2$, with one remaining case: $n=2$, $q\equiv 1(\mathrm{mod}4)$, and $q$ not a square. We settle this case by proving that every positivity preserver on $M_2({\mathbb{F}}_q)$ is injective on the set ${\mathbb{F}}_q^{+}$ of nonzero squares whenever $q\equiv 1(\mathrm{mod}4)$. The proof combines an idempotent reduction of positivity preservers with a well-known property of quadratic characters. This yields the complete classification of entrywise positivity preservers over every finite field and in every fixed dimension.

4 pages; latex