On the classifications of unitary matrices (Q902234)

The main result obtained by the authors is the following theorem: Let \(A \in \text{SU}(p, q)\). Let \(R(\chi_A, {\chi'}_A)\) denote the resultant of the characteristic polynomial \(\chi_A(X)\) and its first derivative \({\chi'}_A(X)\). Then for \(m\geq0\), we have the following. {\parindent=8mm \begin{itemize} \item[(i)] A is regular \(2m\)-loxodromic if and only if \(R(\chi_A, {\chi}'_A ) > 0\). \item [(ii)] A is regular \((2m + 1)\)-loxodromic if and only if \(R(\chi_A, {\chi}'_A ) < 0\). \item [(iii)] A has a repeated eigenvalue if and only if \(R(\chi_A, {\chi}'_A ) = 0\). \end{itemize}} The authors apply their main result to several geometrically interesting situations. They include a detailed discussion of the case \(p+q=4\), giving special attention to the case \(p=3\) and \(q=1\). The authors have a new approach to the set of points defined by \(R(\chi_A, {\chi}'_A ) = 0\), a 3-dimensional subspace which was called the holy grail by \textit{T. Poston} and \textit{I. Stewart} [Geom. Dedicata 27, No. 3, 263--280 (1988; Zbl 0662.58001)]. They indicate that their treatment may also work in higher dimensions. The authors extend Goldman's classification of matrices in \(\mathrm{SU}(2,1)\); they discuss the anti-de Sitter space.