Counting Matrices by Drazin Index

Abstract

Let $F_q^{n \times n} $ denote the algebra of $n \times n$ matrices over $F_q $, the finite field of q elements and for each $A \in F_q^{n \times n} $, let ${\operatorname{Ind}}\,( A )$ denote the Drazin index of A; i.e., ${\operatorname{Ind}}\, ( A )$ is the least nonnegative integer k such that the system of matrix equations (i) $A^{k + 1} X = A^k $, (ii) $XAX = X$ and (iii) $AX = XA$ has a (necessarily unique) solution. This paper determines for each $k\geqq 0$ the number of matrices $A \in F_q^{n \times n} $ with ${\operatorname{Ind}}\, ( A ) = k$. These results are then extended to cover a more general class of finite rings including the ring $Z/Zm$ of integers modulo m.