Simultaneous symplectic spectral decomposition of positive semidefinite matrices
Rudra R. Kamat, Hemant K. Mishra
TL;DR
The paper establishes necessary and sufficient conditions for simultaneous symplectic spectral decomposition of a family of real positive semidefinite matrices with symplectic kernels, showing that such simultaneous diagonalization by a common symplectic matrix occurs precisely when the matrices symplectically commute and their kernel intersection is symplectic. This yields a powerful generalization of Williamson's theorem to families and provides concrete corollaries, including orthosymplectic diagonalization criteria. The results are applied to Gaussian quantum information, giving a criterion for normal-mode decomposition under a common Gaussian unitary, and to statistical mechanics, deriving an analytic partition function for quadratic Hamiltonians. The work has potential impact in areas requiring joint normal-mode analysis under symplectic structure, such as multivariate Gaussian states and quadratic Hamiltonian systems.
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.