Abstract
We determine the number of nilpotent matrices of order $n$ over $\mathbb{F}_q$ that are self-adjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices.
Highlights
Consider matrices of order n over the finite field Fq
We determine the number of nilpotent matrices of order n over Fq that are selfadjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices
For nonsingular linear transformations A the forms given by the matrices G and A GA lead to the same number of self-adjoint N . (When GN is symmetric, so is A GN A, so that A−1N A is self-adjoint for A GA.) if we are interested in the number of self-adjoint nilpotent N, we need only look at g up to scaling and congruence
Summary
Consider matrices of order n over the finite field Fq. Trivially, the total number of such matrices is qn, of which qn(n+1)/2 are symmetric. The number of nilpotent matrices is qn(n−1)—see below for references and yet another proof. The aim of this note is to count symmetric nilpotent matrices, and more generally nilpotent matrices that are self-adjoint for a given nondegenerate symmetric bilinear form
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