Sums and products of symplectic eigenvalues

TL;DR

This paper investigates the properties of symplectic eigenvalues of positive definite matrices, establishing extremal principles and inequalities that extend classical results to the symplectic setting.

Contribution

It introduces analogues of Wielandt's extremal principle and Lidskii's inequalities for symplectic eigenvalues, expanding the theoretical framework.

Findings

Derived extremal principles for symplectic eigenvalues.

Established multiplicative inequalities analogous to Lidskii's.

Extended classical eigenvalue inequalities to symplectic matrices.

Abstract

For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The numbers $d_1(A),\ldots,d_n(A)$ are called the symplectic eigenvalues of $A.$ We derive analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities for symplectic eigenvalues.