Involutary expressions for elements in \(\text{GL}_ n(\mathbb{Z})\) and \(\text{SL}_ n(\mathbb{Z})\)
Let \(\mathbb{Z}\) be the ring of integers. Consider all matrices with entries in \(\mathbb{Z}\). Let \(\text{GL}_ n (\mathbb{Z})\) and \(\text{SL}_ n (\mathbb{Z})\) be the sets of matrices with determinants \(\pm 1\) and 1, respectively. The author shows that every matrix in \(\text{GL}_ n (\mathbb{Z})\) is a product of \(3n+9\) or fewer involutions and that every matrix in \(\text{SL}_ n (\mathbb{Z})\) is a product of \(3n + 11\) or fewer involutions in \(\text{SL}_ n (\mathbb{Z})\). The proof uses imaginative matrix calculations and an application of Dirichlet's theorem.
scientific article; zbMATH DE number 757614