Powers of low rank sparse matrices

Abstract

Let A be a matrix of order n over an arbitrary field F, and let k be a positive integer. We present an algorithm that computes $A^k$ which is faster than the known methods whenever A is sufficiently sparse and its rank is sufficiently small. In fact, already when $rank\left(A\right)=O\left(n^{1-ϵ}\right)$ for a fixed $ϵ>0$ our method yields improved running times.