Enumerating partial linear transformations in a similarity class

Abstract

Let V be a finite-dimensional vector space over the finite field Fq and suppose W and W are subspaces of V . Two linear transformations T : W → V and T : W → V are said to be similar if there exists a linear isomorphism S : V → V with SW = W such that S • T = T • S. Given a linear map T defined on a subspace W of V , we give an explicit formula for the number of linear maps that are similar to T . Our results extend a theorem of Philip Hall that settles the case W = V where the above problem is equivalent to counting the number of square matrices over Fq in a conjugacy class.