The asymptotic complexity of matrix reduction over finite fields

Consider an invertible n × n matrix over some field. The Gauss-Jordan elimination reduces this matrix to the identity matrix using at most n row operations and in general that many operations might be needed. In [1] the authors considered matrices in GL(n, q), the set of n × n invertible matrices in the finite field of q elements, and provided an algorithm using only row operations which performs asymptotically better than the Gauss-Jordan elimination. More specifically their ‘striped elimination algorithm’ has asymptotic complexity n 2 logq n . Furthermore they proved that up to a constant factor this algorithm is best possible as almost all matrices in GL(n, q) need asymptotically at least n 2 2 logq n operations. In this short note we show that the ‘striped elimination algorithm’ is asymptotically optimal by proving that almost all matrices in GL(n, q) need asymptotically at least n 2 logq n operations.