The probability that an operator is nilpotent

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Abstract

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
Number of pages5
JournalAmerican mathematical monthly
Volume128
Issue number4
DOIs
Publication statusAccepted/In press - 17 Sep 2020

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