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These are the matrices considered to be `good' in the Holt-Rees version of Norton's irreducibility test in the MeatAxe algorithm. We prove that, for any finite field $\\mathbb{F}_q$, the proportion of matrices in ${\\rm M}(n,\\mathbb{F}_q)$ that are `not good' decays exponentially to zero as the dimension $n$ approaches infinity. 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Praeger, S.P. Glasby","submitted_at":"2014-05-22T05:29:01Z","abstract_excerpt":"An element $X$ in the algebra ${\\rm M}(n,\\mathbb{F})$ of all $n\\times n$ matrices over a field $\\mathbb{F}$ is said to be $f$-cyclic if the underlying vector space considered as an $\\mathbb{F}[X]$-module has at least one cyclic primary component. These are the matrices considered to be `good' in the Holt-Rees version of Norton's irreducibility test in the MeatAxe algorithm. We prove that, for any finite field $\\mathbb{F}_q$, the proportion of matrices in ${\\rm M}(n,\\mathbb{F}_q)$ that are `not good' decays exponentially to zero as the dimension $n$ approaches infinity. 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