Let $I_n$ be the $n\times n$ identity matrix and $J=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]$. A matrix A is called symplectic if $A^{T}JA=J$. A symplectic matrix A is a commutator of symplectic involutions if $A=XY{X}^{-1}{Y}^{-1}$, where X and Y are symplectic matrices satisfying $X^2=Y^2=I$. In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions.

中文翻译：

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辛对合交换子的乘积

让${我}_n$成为$n\times n$单位矩阵和$Ĵ=\left[\begin{array}{cc}0& {我}_n\\ -{我}_n& 0\end{array}\right]$. 如果矩阵A称为辛矩阵${\mathrm{一个}}^{吨}Ĵ\mathrm{一个}=Ĵ$. 一个辛矩阵A是一个辛对合的交换子，如果$\mathrm{一个}=X是X^{-1}{是}^{-1}$, 其中X和Y是辛矩阵满足$X^2={是}^2=我$. 在本文中，我们给出了将复数域上的辛矩阵表示为两个辛对合交换子的乘积的充要条件。