Simultaneous symplectic spectral decomposition of positive semidefinite matrices

TL;DR

This paper characterizes conditions for simultaneously decomposing multiple positive semidefinite matrices using symplectic spectral methods, extending known results to broader classes.

Contribution

It provides necessary and sufficient conditions for symplectic spectral decomposition of families of positive semidefinite matrices, generalizing previous results.

Findings

Established algebraic conditions for symplectic spectral decomposition.

Generalized orthosymplectic spectral diagonalization to positive semidefinite matrices.

Extended known results from positive definite to positive semidefinite matrices.

Abstract

We establish necessary and sufficient conditions on simultaneous symplectic spectral decomposition of a family of $2n \times 2n$ real positive semidefinite matrices with symplectic kernels. We also provide a precise algebraic condition on a $2n \times 2n$ real positive semidefinite matrix with symplectic kernel for orthosymplectic spectral diagonalization, which generalizes a known result for positive definite matrices.