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Derangements and eigenvalue-free elements in finite classical groups. (English) Zbl 0936.15020
One of the questions addressed in this paper is: given the matrix ring $$M(n,q)$$ of $$n\times n$$ matrices over the field $$F_q$$ with $$q$$ elements, what is the probability $$\upsilon(M;n,q)$$ that a randomly chosen matrix has no eigenvalues? Regarding different $$M$$, the authors derive asymptotic expressions for $$\upsilon$$, in the limit $$n\to\infty$$, that has as the leading term $$e^{-1}$$.

##### MSC:
 15B52 Random matrices (algebraic aspects) 20G40 Linear algebraic groups over finite fields
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