Williamson theorem


In the context of linear algebra and symplectic geometry, the Williamson theorem concerns the diagonalization of positive definite matrices through symplectic matrices.
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.

Proof

The derivation of the result hinges on a few basic observations:
  1. The real matrix, with, is well-defined and skew-symmetric.
  2. For any invertible skew-symmetric real matrix, there is such that, where a real positive-definite diagonal matrix containing the singular values of.
  3. For any orthogonal, the matrix is such that.
  4. If diagonalizes, meaning it satisfies then is such that Therefore, taking, the matrix is also a symplectic matrix, satisfying.