Simultaneous symplectic spectral decomposition of positive semidefinite matrices

Abstract

We establish necessary and sufficient conditions on simultaneous symplectic spectral decomposition of a family of $2n\times 2n$ real positive semidefinite matrices with symplectic kernels. We also provide a precise algebraic condition on a $2n\times 2n$ real positive semidefinite matrix with symplectic kernel for orthosymplectic spectral diagonalization, which generalizes a known result for positive definite matrices.

arXiv comment12 pages

References in corpus (6)

• The Real Symplectic Groups in Quantum Mechanics and Optics
• Symplectic eigenvalue problem via trace minimization and Riemannian optimization
• Derivatives of symplectic eigenvalues and a Lidskii type theorem
• Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem
• Real Normal Operators and Williamson's Normal Form
• Quadratic integrals of motion for the systems of identical particles