Nilpotent-independent sets and estimation in matrix algebras

Brian P. Corr, Tomasz Popiel, Cheryl E. Praeger
2015 LMS Journal of Computation and Mathematics  
Efficient methods for computing with matrices over finite fields often involverandomisedalgorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families$N$of elements in the algebra of all$d\times d$matrices over a field of order$q$, where membership of a matrix in$N$depends only
more » ... x in$N$depends only on its 'invertible part'. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on$N$, so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt–ReesMEATAXEalgorithm for testing irreducibility of matrix algebras.
doi:10.1112/s146115701500008x fatcat:qqdh72vt4jd2zkm2scjyl5v7g4