Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem

TL;DR

This paper extends the concept of symplectic eigenvalues to positive-semidefinite matrices by generalizing Williamson's diagonal form and proves a trace minimization theorem in this broader context.

Contribution

It introduces a generalized Williamson's diagonal form for positive-semidefinite matrices and establishes the trace minimization theorem for symplectic eigenvalues in this setting.

Findings

Generalized Williamson's diagonal form for positive-semidefinite matrices

Defined symplectic eigenvalues for this broader class

Proved the trace minimization theorem in the new setting

Abstract

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.