Majorization between symplectic spectra of positive semidefinite matrices

TL;DR

This paper explores the relationships between symplectic spectra of positive semidefinite matrices, establishing majorization and weak supermajorization relations, and characterizing the convex hull of symplectic orbits.

Contribution

It introduces new majorization relations between symplectic spectra and characterizes the convex hull of symplectic orbits of positive semidefinite matrices.

Findings

Majorization of symplectic spectra implies inclusion in the convex hull of symplectic orbit.

Weak supermajorization is a necessary condition for such inclusion.

Connections to doubly stochastic and symplectic matrices underpin the results.

Abstract

Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.