On the ϕJ polar decomposition of matrices

We present new results on the ϕJ polar decomposition of matrices. We show that every symplectic matrix may be written as a product of symplectic operation matrices. We present a simple form attained under symplectic equivalence, which makes it easy to determine if a matrix does not have a ϕJ polar decomposition. We also determine the rank 4 matrices with ϕJ polar decomposition.