Counting matrices over a finite field with all eigenvalues in the field

Abstract

Given a finite field $\mathbb{F}$ and a positive integer $n$, we give a procedure to count the $n\times n$ matrices with entries in $\mathbb{F}$ with all eigenvalues in the field. We give an exact value for any field for values of $n$ up to $4$, and prove that for fixed $n$, as the size of the field increases, the proportion of matrices with all eigenvalues in the field approaches $1∕n!$. As a corollary, we show that for large fields almost all matrices with all eigenvalues in the field have all eigenvalues distinct. The proofs of these results rely on the fact that any matrix with all eigenvalues in $\mathbb{F}$ is similar to a matrix in Jordan canonical form, and so we proceed by enumerating the number of $n\times n$ Jordan forms, and counting how many matrices are similar to each one. A key step in the calculation is to characterize the matrices that commute with a given Jordan form and count how many of them are invertible.