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 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:

  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.
  2. 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.
  3. 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.
  4. 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". American Journal of Mathematics. 58 (1): 141–163. doi:10.2307/2371062. ISSN 0002-9327. JSTOR 2371062
  2. ^ Nicacio, F. (2021-12-01). "Williamson theorem in classical, quantum, and statistical physics". American Journal of Physics. 89 (12): 1139–1151. arXiv:2106.11965. Bibcode:2021AmJPh..89.1139N. doi:10.1119/10.0005944. ISSN 0002-9505
  3. ^ Yusofsani, Mohammad (25 November 2018). "Symplectic Geometry and Wiliamson's Theorem". Retrieved 25 November 2018.