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In the context of linear algebra and symplectic geometry, the Williamson theorem concerns the diagonalization of positive definite matrices through symplectic matrices. [1] [2] [3]
More precisely, given a strictly positive-definite Hermitian real matrix , the theorem ensures the existence of a real symplectic matrix , and a diagonal positive real matrix , such that where denotes the 2x2 identity matrix.
The derivation of the result hinges on a few basic observations: