Decomposition of matrices into three involutions (Q1112130)

The author shows that every complex \(n\times n\) matrix A with determinant \(\pm 1\) and \(\dim \ker (A-\alpha)\leq [n/2]\) for any \(\alpha\in {\mathbb{C}}\) is the product of three involutions. Furthermore, a necessary condition for the decomposability of a complex \(n\times n\) matrix A into the product of three involutions is given: Let \(m=\dim \ker (A-\beta)\) and \(r=\dim \ker (A-\beta^{-3})\) for \(\beta\in {\mathbb{C}}\). Then \(m\leq (2n+r)/3\) and \(m\leq [3n/4]\) for any \(\beta\in {\mathbb{C}}\), \(\beta\neq 0\), and \(\beta^ 4\neq 1\). Using these results the author is able to characterize complex \(5\times 5\) matrices which are products of three involutions.