Optimal unit triangular factorization of symplectic matrices

TL;DR

This paper proves that any symplectic matrix can be optimally factored into no more than five unit triangular symplectic matrices, improving previous bounds and enabling new optimization methods on symplectic groups.

Contribution

It establishes the minimal number of factors needed for symplectic matrix factorization and extends results to structured subsets, enhancing theoretical understanding.

Findings

Any symplectic matrix can be factored into at most 5 unit triangular symplectic matrices.

The factorization number 5 is proven to be optimal.

Provides an unconstrained optimization method on the symplectic group.

Abstract

We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing triangular factorization of symplectic matrices which gives proof of 9 factors. We also show the corresponding improved conclusions for structured subsets of symplectic matrices. This factorization further provides an unconstrained optimization method on $2d$-by-$2d$ real symplectic group (a $2d^2+d$-dimensional Lie group) with $2d^2+3d$ parameters.