On the classifications of unitary matrices

Abstract

We classify the dynamical action of matrices in $\mathrm{SU}(p, q)$ using the coefficients of their characteristic polynomial. This generalises earlier work of Goldman for $\mathrm{SU}(2, 1)$ and the classical result for $\mathrm{SU}(1, 1)$, which is conjugate to $\mathrm{SL}(2, \mathbb{R})$. As geometrical applications, we show how this enables us to classify automorphisms of real and complex hyperbolic space and anti de Sitter space.