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arXiv · 1803.04647

On symplectic eigenvalues of positive definite matrices

Rajendra Bhatia·Tanvi Jain
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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.

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Rajendra Bhatia, Tanvi Jain. 2018-03-13. On symplectic eigenvalues of positive definite matrices. https://doi.org/10.1063/1.4935852

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