The probability that an operator is nilpotent

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Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.
Original languageEnglish
Pages (from-to)371-375
Number of pages5
JournalThe American Mathematical Monthly
Issue number4
Early online date23 Mar 2021
Publication statusPublished - 30 Apr 2021


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