The Meat-axe and f-cyclic matrices

Let M(d,F) denote the algebra of d×d matrices over a field F, and denote by mX(t) and cX(t) the minimal and the characteristic polynomials of X∈M(d,F). We call X an f-cyclic matrix if f is an irreducible factor of mX(t) which does not divide cX(t)/mX(t). We present a version of the Meat-axe algorithm that uses f-cyclic matrices. One advantage of f-cyclic matrices is that they unify and generalize previous work of Parker, Holt and Rees, Ivanyos and Lux, Neumann and Praeger. The greater abundance of f-cyclic matrices may lead to an improved probability/complexity analysis of the Meat-axe. The difficulties that occur when the Schur index exceeds one are explored.