Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem

Abstract

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.

9 pages

References in corpus (1)

• Algebraic canonical quantization of lumped superconducting networks

Cited by in corpus (5)

• Block perturbation of symplectic matrices in Williamson's theorem
• An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators
• Configurable photonic simulator for quantum field dynamics
• A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold
• Simultaneous symplectic spectral decomposition of positive semidefinite matrices