Each symplectic matrix is a product of four symplectic involutions

Abstract

Gustafson, Halmos, and Radjavi in 1973 proved that each matrix A with det⁡A=±1 is a product of four involutions. We prove that these involutions can be taken to be symplectic if A is symplectic (every symplectic matrix has unit determinant). Using this result we give an alternative proof of Laffey's theorem that every nonsingular even size matrix is a product of skew symmetric matrices. Ballantine in 1978 proved that each matrix A with |det⁡A|=1 is a product of four coninvolutions. We prove that these coninvolutions can be taken to be symplectic if A is symplectic. We also prove that each Hamiltonian matrix is a sum of two square zero Hamiltonian matrices.