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ABSTRACT: A matrix is said to be {\it cyclic} if its characteristic polynomial is equal
to its minimal polynomial. Cyclic matrices play an important role in some
algorithms for matrix group computation, such as the Cyclic Meataxe developed
by P. M. Neumann and C. E. Praeger in 1999. In that year also, G. E. Wall and
J. E. Fulman independently found the limiting proportion of cyclic matrices in
general linear groups over a finite field of fixed order q as the dimension n
approaches infinity, namely $(1-q^{-5}) \prod_{i=3}^\infty (1-q^{-i}) = 1 -
q^{-3} + O(q^{-4}).$ We study cyclic matrices in a maximal reducible matrix
group or algebra, that is, in the largest subgroup or subalgebra that leaves
invariant some proper nontrivial subspace. We modify Wall's generating function
approach to determine the limiting proportions of cyclic matrices in maximal
reducible matrix groups and algebras over a field of order q, as the dimension
of the underlying vector space increases while that of the invariant subspace
remains fixed. The limiting proportion in a maximal reducible group is proved
to be $1 - q^{-2} + O(q^{-3})$; note the change of the exponent of q in the
second term of the expansion. Moreover, we exhibit in each maximal reducible
matrix group a family of noncyclic matrices whose proportion is $q^{-2} +
O(q^{-3})$.
05/2011;
