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 .

References