Stein.s method and the rank distribution of random matrices over finite fields

With Qq,n the distribution of n minus the rank of a matrix chosen uniformly from the collection of all n.(n+m) matrices over the finite field Fq of size q.2, and Qq the distributional limit of Qq,n as n.. , we apply Stein.s method to prove the total variation bound 18qn+m+1..Qq,n.Qq.TV.3qn+m+1. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.