Proximal linear maps (Q930806)

Let \(V\) be a finite-dimensional real vector space. A linear endomorphism of \(V\) is called \textit{proximal} if it has exactly one eigenvalue of maximal absolute value. Proximal maps had been used in many contexts. The main purpose of the present expository paper is to popularize the notion and to draw attention to the following theorem of Gol'dsheid and Margulis. Let \(G\) be a strongly irreducible subgroup of \(GL(V)\) containing a proximal map. Then every Zariski dense subgroup of \(G\) is strongly irreducible and contains a proximal map.