The number of linear transformations defined on a subspace with given invariant factors

TL;DR

This paper derives an explicit formula for counting linear transformations with prescribed invariant factors from a subspace to a vector space over a finite field, extending previous results and applying to control theory and matrix enumeration.

Contribution

It provides the first explicit counting formula for linear transformations with given invariant factors from a subspace to a vector space, generalizing prior work for the case when the subspace equals the entire space.

Findings

Derived an explicit formula for the count of such transformations.

Extended the Gerstenhaber-Reiner formula for matrices with a given characteristic polynomial.

Provided new proofs for recent results in linear control theory.

Abstract

Given a finite-dimensional vector space $V$ over the finite field $\mathbb{F}_q$ and a subspace $W$ of $V$, we consider the problem of counting linear transformations $T:W\to V$ which have prescribed invariant factors. The case $W=V$ is a well-studied problem that is essentially equivalent to counting the number of square matrices over $\mathbb{F}_q$ in a conjugacy class and an explicit formula is known in this case. On the other hand, the case of general $W$ is also an interesting problem and there hasn't been substantive progress in this case for over two decades, barring a special case where all the invariant factors of $T$ are of degree zero. We extend this result to the case of arbitrary $W$ by giving an explicit counting formula. As an application of our results, we give new proofs of some recent enumerative results in linear control theory and derive an extension of the Gerstenhaber-Reiner formula for the number of square matrices over $\mathbb{F}_q$ with given characteristic polynomial.