The products of involutions in a matrix centralizer

Abstract

A square matrix $A$ is an involution if $A^{2} = I$. The centralizer of a square matrix $S$ denoted by $\mathscr{C}(S)$ is the set of all $A$ such that $AS = SA$ over an algebraically closed field of characteristic not equal to 2. We determine necessary and sufficient conditions for $A \in \mathscr{C}(S)$ to be a product of involutions in $\mathscr{C}(S)$ where $S$ is a basic Weyr matrix with homogeneous Weyr structure of length 3. Finally, we will show some results for the case when the length of the Weyr structure is greater than 3.