Derivatives of symplectic eigenvalues and a Lidskii type theorem

Abstract

Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist $n$ positive numbers called the symplectic eigenvalues of $A,$ and a basis of ${\mathbb{R}}^{2n}$ called the symplectic eigenbasis of $A$ corresponding to these numbers. In this paper, we discuss the differentiability (analyticity) of the symplectic eigenvalues and corresponding symplectic eigenbasis for differentiable (analytic) map $t\mapsto A(t),$ and compute their derivatives. We then derive an analogue of Lidskii's theorem for symplectic eigenvalues as an application.

arXiv comment36 pages

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Cited by in corpus (8)

• Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem
• Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric
• Block perturbation of symplectic matrices in Williamson's theorem
• An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators
• Majorization in some symplectic weak supermajorizations
• Equality in some symplectic eigenvalue inequalities
• A Result About the Classification of Quantum Covariance Matrices Based on Their Eigenspectra
• Simultaneous symplectic spectral decomposition of positive semidefinite matrices