Some asymptotics for triangular matrices over finite fields (Q1809101)

The author compares the asymptotic behavior of certain probabilities for upper triangular matrices over the finite field \(F_{q}\) to the asymptotic behavior of the corresponding probabilities for arbitrary matrices. As \(q\rightarrow \infty \), the probability that an \(n\times n\) upper triangular matrix over \(F_{q}\) has rank equal to \(n\) (or being the matrix diagonalizable) approaches \(1\). Similar results are obtained for arbitrary matrices.