On the products of commutators of real $J$-symmetries

Abstract

Let $J_{2n}=\begin{bmatrix} 0&I_n\\-I_n&0\end{bmatrix}$. A 2$n$-by-2$n$ complex matrix $A$ is said to be symplectic if $A^TJA=J$. If $A$ is symplectic and rank$(A-I)=1$, then $A$ is called a $J$-symmetry. It is known that every 2$n$-by-2$n$ complex symplectic matrix can be written as a product of $n+1$ commutators of $J$-symmetries. We consider the real case and study the properties of real $J$-symmetries and commutators of real $J$-symmetries. We prove that if $A$ is a $2n$-by-$2n$ real symplectic matrix, with $\mathrm{rank}(A-I)=m$, then $A$ is a product of $\frac{3m}{2}-2\lfloor \frac{m}{4} \rfloor$ commutators of real $J$-symmetries if $J(A-I)$ is skew-symmetric, and $A$ is a product of $3 \lceil \frac{m}{2} \rceil$ commutators of real $J$-symmetries if $J(A-I)$ is not skew-symmetric.