  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.
