Every real symplectic matrix is a product of real symplectic involutions

Abstract

A 2n-by-2n matrix A is symplectic if AT[0I−I0]A=[0I−I0]. It is known that if n>1, then every 2n-by-2n complex symplectic matrix is a product of four symplectic involutions. We consider the real case. We give an example of a real symplectic matrix which is a product of two complex symplectic involutions but is not a product of two real symplectic involutions. We show that every 4-by-4 real symplectic matrix is a product of four real symplectic involutions. We use this result to show that for n>1, every 2n-by-2n real symplectic matrix is a product of a finite number of real symplectic involutions.