О параметрических представлениях ортогональных и симплектических матриц

Abstract

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.