5 research outputs found
Counting symmetric nilpotent matrices
We determine the number of nilpotent matrices of order n over Fq that are self-adjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices. Keywords: Nilpotent; symmetric; matrix; endomorphism; enumeratio
The probability that an operator is nilpotent
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.Comment: 5 pages, title change
Counting symmetric nilpotent matrices
We determine the number of nilpotent matrices of order n over Fq that are self-adjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices. Keywords: Nilpotent; symmetric; matrix; endomorphism; enumeratio