On symplectic eigenvalues of positive definite matrices

If A is a 2n x 2n real positive definite matrix, then there exists a symplectic matrix M such that M-T AM = [GRAPHICS] where D = diag(d(1)(A), ..., 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 At, between the symplectic eigenvalues of m matrices A(1), ..., A(m) and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues. (C) 2015 AIP Publishing LLC.