Williamson theorem
Appearance
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 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
[edit]The derivation of the result hinges on a few basic observations:
- The real matrix , with , is well-defined and skew-symmetric.
- Any skew-symmetric real matrix can be block-diagonalized via orthogonal real matrices, meaning there is such that with a real positive-definite diagonal matrix containing the singular values of .
- For any orthogonal , the matrix is such that .
- If diagonalizes , meaning it satisfies then is such that Therefore, taking , the matrix is also a symplectic matrix, satisfying .
See also
[edit]References
[edit]- ^ 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.
- ^ Nicacio, F. (2021-12-01). "Williamson theorem in classical, quantum, and statistical physics". American Journal of Physics. 89 (12): 1139–1151. arXiv:2106.11965. doi:10.1119/10.0005944. ISSN 0002-9505.
- ^ Yusofsani, Mohammad (25 November 2018). "Symplectic Geometry and Wiliamson's Theorem" (PDF). Retrieved 25 November 2018.