Williamson theorem

In the context of linear algebra and symplectic geometry, the Williamson theorem concerns the diagonalization of positive definite matrices through symplectic matrices.{{Cite journal |last=Williamson |first=John |date=1936 |title=On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems |url=https://www.jstor.org/stable/2371062 |journal=American Journal of Mathematics |volume=58 |issue=1 |pages=141–163 |doi=10.2307/2371062 |jstor=2371062 |issn=0002-9327|url-access=subscription }}{{Cite journal |last=Nicacio |first=F. |date=2021-12-01 |title=Williamson theorem in classical, quantum, and statistical physics |journal=American Journal of Physics |volume=89 |issue=12 |pages=1139–1151 |doi=10.1119/10.0005944 |issn=0002-9505|arxiv=2106.11965 |bibcode=2021AmJPh..89.1139N }}{{Cite web |last=Yusofsani |first=Mohammad |date=25 November 2018 |title=Symplectic Geometry and Wiliamson's Theorem |url=https://math.arizona.edu/~rsims/ma541/Seye_lec.pdf |access-date=25 November 2018}}

More precisely, given a strictly positive-definite $2n\times 2n$ Hermitian real matrix $M\in\mathbb{R}^{2n\times 2n}$ , the theorem ensures the existence of a real symplectic matrix $S\in\mathbf{Sp}(2n,\mathbb{R})$ , and a diagonal positive real matrix $D\in\mathbb{R}^{n\times n}$ , such that $SMS^T = I_2\otimes D \equiv D\oplus D,$ where $I_2$ denotes the 2x2 identity matrix.

Proof

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

The real matrix $M^{-1/2} (J\otimes I_n) M^{-1/2}$ , with $J\equiv\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ , is well-defined and skew-symmetric.
For any invertible skew-symmetric real matrix $A\in\mathbb{R}^{2n\times 2n}$ , there is $O\in\mathbf{O}(2n)$ such that $OAO^T= J\otimes \Lambda$ , where $\Lambda$ a real positive-definite diagonal matrix containing the singular values of $A$ .
For any orthogonal $O\in\mathbf O(2n)$ , the matrix $S= \left(I_2\otimes\sqrt D\right)O M^{-1/2}$ is such that $SMS^T=J\otimes D$ .
If $O\in\mathbf O(2n)$ diagonalizes $M^{-1/2} (J\otimes I_n) M^{-1/2}$ , meaning it satisfies $OM^{-1/2} (J\otimes I_n) M^{-1/2}O^T=J\otimes\Lambda,$ then $S= \left(I_2\otimes\sqrt D\right)O M^{-1/2}$ is such that $S(J\otimes I_n)S^T=J\otimes (D\Lambda) .$ Therefore, taking $D=\Lambda^{-1}$ , the matrix $S$ is also a symplectic matrix, satisfying $S(J\otimes I_n)S^T=J\otimes I_n$ .

References

Category:Theorems in linear algebra

Category:Matrices (mathematics)

Category:Symplectic geometry