Unit triangular factorization of the matrix symplectic group

Understand

In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices.

• This structure-preserving factorization of the symplectic matrices immediately reveals two well-known features that, (i) the determinant of any symplectic matrix is one, (ii) the matrix symplectic group is path connected, as well as a new feature that (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group.
• 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 problems with symplectic constraints under certain circumstances.