On symplectic eigenvalues of positive definite matrices

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 $A\mathrm{.}$ 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 $A_1,\dots ,A_m$ and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.