On symplectic eigenvalues of positive definite matrices

TL;DR

This paper investigates fundamental inequalities involving symplectic eigenvalues of positive definite matrices, including relations, perturbation results, and comparisons with ordinary eigenvalues, advancing understanding in symplectic matrix analysis.

Contribution

It derives new inequalities and principles related to symplectic eigenvalues, including their relations, perturbation bounds, and comparisons with standard eigenvalues.

Findings

Relations between symplectic eigenvalues of A and A^t

Inequalities involving symplectic eigenvalues of multiple matrices and their Riemannian mean

Perturbation theorems and variational principles for symplectic eigenvalues

Abstract

If $A$ is a $2n \times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^TAM = \left [ \begin{array}{cc} D & O \\ O & D \end{array} \right ]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of $A.$ In this paper we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of $A$ and those of $A^t,$ between the symplectic eigenvalues of $m$ matrices $A_1, \ldots, A_m$ and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.