English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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.

Unit Triangular Factorization of the Matrix Symplectic Group