Williamson theorem
In the context of linear algebra and symplectic geometry, the Williamson theorem concerns the diagonalization of positive definite matrices through symplectic matrices. More precisely, given a strictly positive-definite 2 n × 2 n {\displaystyle 2n\times 2n} Hermitian real matrix M ∈ R 2 n × 2 n {\displaystyle M\in \mathbb {R} ^{2n\times 2n}} , the theorem ensures the existence of a real symplectic matrix S ∈ S p ( 2 n , R ) {\displaystyle S\in \mathbf {Sp} (2n,\mathbb {R} )} , and a diagonal positive real matrix D ∈ R n × n {\displaystyle D\in \mathbb {R} ^{n\times n}} , such that S M S T = I 2 ⊗ D ≡ D ⊕ D , {\displaystyle SMS^{T}=I_{2}\otimes D\equiv D\oplus D,} where I 2 {\displaystyle I_{2}} denotes the 2x2 identity matrix.