# Williamson theorem

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In the context of [linear algebra](/source/Linear_algebra) and [symplectic geometry](/source/Symplectic_geometry), the **Williamson theorem** concerns the diagonalization of [positive definite matrices](/source/Definite_matrix) through [symplectic matrices](/source/Symplectic_matrix).[1][2][3]

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](/source/Identity_matrix).

## Proof

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

1. 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.
1. 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](/source/Diagonal_matrix) containing the [singular values](/source/Singular_value) of A.
1. 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=I_2\otimes D.
1. 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

1. Williamson, John (1936). ["On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems"](https://www.jstor.org/stable/2371062). *American Journal of Mathematics*. **58** (1): 141–163. [doi:10.2307/2371062](https://doi.org/10.2307/2371062). [ISSN 0002-9327](https://www.worldcat.org/issn/0002-9327). [JSTOR 2371062](https://www.jstor.org/stable/2371062)

1. Nicacio, F. (2021-12-01). "Williamson theorem in classical, quantum, and statistical physics". *American Journal of Physics*. **89** (12): 1139–1151. [arXiv:2106.11965](https://arxiv.org/abs/2106.11965). [Bibcode:2021AmJPh..89.1139N](https://ui.adsabs.harvard.edu/abs/2021AmJPh..89.1139N). [doi:10.1119/10.0005944](https://doi.org/10.1119/10.0005944). [ISSN 0002-9505](https://www.worldcat.org/issn/0002-9505)

1. Yusofsani, Mohammad (25 November 2018). ["Symplectic Geometry and Wiliamson's Theorem"](https://math.arizona.edu/~rsims/ma541/Seye_lec.pdf). Retrieved 25 November 2018.

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Adapted from the Wikipedia article [Williamson theorem](https://en.wikipedia.org/wiki/Williamson_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Williamson_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.