Every 2n-by-2n complex symplectic matrix is a product of n + 1 symplectic dilatations

Abstract

A $2n\times 2n$ complex matrix A is symplectic if $A^{T}JA=J$ where $J=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]$. We say that A is a symplectic dilatation if A is symplectic and is similar to $\left[a\right]\oplus \left[a^{-1}\right]\oplus I_{2n-2}$. If $n>1$, we show that every $2n\times 2n$ complex symplectic matrix A is a product of n symplectic dilatations except when A is similar to $J_2(-1)\oplus -I_{2n-2}$, in which case A is a product of $n+1$ symplectic dilatations.