Let ${ \rm M}(V)={ \rm M}(n,\mathbb{ F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{ F}_q$, and let ${ \rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\mathbb{ F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${ \rm M}(V)_U$ is at least $q^{ -2}\left(1+c_1q^{ -1}\right)$, and at most $q^{ -2}\left(1+c_2q^{ -1}\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\frac43$ and $c_2=\frac{ 35}3$ suffice.
Categories: math.RA math.RT