In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structured preserving factorization of the symplectic matrices immediately reveals several important inferences, such as, (\romannumeral1) the determinant of symplectic matrix is one, (\romannumeral2) the matrix symplectic group is path connected, (\romannumeral3) all the unit triangular symplectic matrices forms a set of generators of the matrix symplectic group, (\romannumeral4) the $2d$-by-$2d$ matrix symplectic group is a smooth manifold of dimension $2d^{2}+d$. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the matrix symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problem with symplectic constraints.

中文翻译：

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矩阵辛群的单位三角分解

在这项工作中，我们证明了任何辛矩阵都可以分解为不超过 9 个单位的三角辛矩阵。辛矩阵的这种结构化保留分解立即揭示了几个重要的推论，例如，(\romannumeral1) 辛矩阵的行列式是一个，(\romannumeral2) 矩阵辛群是路径连通的，(\romannumeral3) 所有单位三角辛矩阵构成矩阵辛群的一组生成元，(\romannumeral4) $2d$-by-$2d$ 矩阵辛群是维度$2d^{2}+d$ 的光滑流形。此外，这种分解为矩阵辛群及其结构化子集的无约束参数化提供了有效的方法。