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added paragraph about Williamson theorem, which ensures "real diagonalisation" via symplectic matrices |
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The Cholesky decomposition is especially useful for efficient numerical calculations.
A closely related decomposition is the [[Cholesky decomposition#LDL decomposition|LDL decomposition]], <math>\ M = L D L^*\ ,</math> where <math>D</math> is diagonal and <math>\ L\ </math> is [[Triangular matrix#Unitriangular matrix|lower unitriangular]].
===Williamson theorem===
Any <math>2n\times 2n </math> positive definite Hermitian real matrix <math>M </math> can be diagonalized via symplectic (real) matrices. More precisely, [[Williamson theorem|Williamson's theorem]] ensures the existence of symplectic <math>S\in\mathbf{Sp}(2n,\mathbb{R}) </math> and diagonal real positive <math>D\in\mathbb{R}^{n\times n} </math> such that <math>SMS^T=D\oplus D </math>.
== Other characterizations ==
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