Symplectic matrices are subject to certain conditions that are inherent to the Jacobian matrices of transformations preserving the Hamiltonian form of differential equations. A formula is derived which parameterizes symplectic matrices by symmetric matrices. An analogy is drawn between the obtained formula and the Cayley formula that connects orthogonal and antisymmetric matrices. It is shown that orthogonal and antisymmetric matrices are transformed by the covariant law when replacing the Cartesian coordinate system. Similarly, the covariance of transformations of symplectic and symmetric matrices is proved.From Cayley formulas and their analog, a series of matrix relations is obtained which connect orthogonal and symmetric matrices, together with similar relations connecting symplectic and symmetric matrices.

中文翻译：

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正交辛辛矩阵的参数表示

辛矩阵要服从某些条件，这些条件对于保留雅各布的哈密顿形式的微分方程的变换的雅可比矩阵是固有的。得出一个通过对称矩阵对辛矩阵进行参数化的公式。在获得的公式与连接正交矩阵和反对称矩阵的Cayley公式之间进行类推。结果表明，当替换笛卡尔坐标系时，正交矩阵和反对称矩阵通过协变量定律进行变换。类似地，证明了辛对称矩阵的变换的协方差。从Cayley公式及其类似物中，获得了一系列连接正交矩阵和对称矩阵的矩阵关系，以及连接辛对称矩阵的相似关系。