Williamson theorem

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 ${\displaystyle 2n\times 2n}$ Hermitian real matrix ${\displaystyle M\in \mathbb {R} ^{2n\times 2n}}$, the theorem ensures the existence of a real symplectic matrix ${\displaystyle S\in \mathbf {Sp} (2n,\mathbb {R} )}$, and a diagonal positive real matrix ${\displaystyle D\in \mathbb {R} ^{n\times n}}$, such that $${\displaystyle SMS^{T}=I_{2}\otimes D\equiv D\oplus D,}$$where ${\displaystyle I_{2}}$ denotes the 2x2 identity matrix.

Proof

The derivation of the result hinges on a few basic observations:

1. The real matrix ${\displaystyle M^{-1/2}(J\otimes I_{n})M^{-1/2}}$, with ${\displaystyle J\equiv {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$, is well-defined and skew-symmetric.
2. For any invertible skew-symmetric real matrix ${\displaystyle A\in \mathbb {R} ^{2n\times 2n}}$, there is ${\displaystyle O\in \mathbf {O} (2n)}$ such that ${\displaystyle OAO^{T}=J\otimes \Lambda }$, where ${\displaystyle \Lambda }$ a real positive-definite diagonal matrix containing the singular values of ${\displaystyle A}$.
3. For any orthogonal ${\displaystyle O\in \mathbf {O} (2n)}$, the matrix ${\displaystyle S=\left(I_{2}\otimes {\sqrt {D}}\right)OM^{-1/2}}$ is such that ${\displaystyle SMS^{T}=J\otimes D}$.
4. If ${\displaystyle O\in \mathbf {O} (2n)}$ diagonalizes ${\displaystyle M^{-1/2}(J\otimes I_{n})M^{-1/2}}$, meaning it satisfies $${\displaystyle OM^{-1/2}(J\otimes I_{n})M^{-1/2}O^{T}=J\otimes \Lambda ,}$$then ${\displaystyle S=\left(I_{2}\otimes {\sqrt {D}}\right)OM^{-1/2}}$ is such that $${\displaystyle S(J\otimes I_{n})S^{T}=J\otimes (D\Lambda ).}$$Therefore, taking ${\displaystyle D=\Lambda ^{-1}}$, the matrix ${\displaystyle S}$ is also a symplectic matrix, satisfying ${\displaystyle S(J\otimes I_{n})S^{T}=J\otimes I_{n}}$.