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→Cholesky decomposition: Fixing formula for Cholesky decomposition |
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* Applicable to: square, [[Positive-definite matrix|positive-definite]] real matrix ''A'' with order 2''n''×2''n''.
* Decomposition: <math>A=S^\mathsf{T}\operatorname{diag}(D,D)S</math>, where <math>S \in \text{Sp}(2n)</math> is a [[symplectic matrix]] and ''D'' is a nonnegative ''n''-by-''n'' diagonal matrix.<ref>{{Cite journal|last=Idel|first=Martin|last2=Soto Gaona|first2=Sebastián|last3=Wolf|first3=Michael M.|date=2017-07-15|title=Perturbation bounds for Williamson's symplectic normal form|journal=Linear Algebra and Its Applications|volume=525|pages=45–58|doi=10.1016/j.laa.2017.03.013|arxiv=1609.01338}}</ref>
===Matrix square root===
{{main|Square root of a matrix}}
* Decomposition: <math>A=BB</math>, not unique in general.
* In the case of positive semidefinite <math>A</math>, there is a unique positive semidefinite <math>B</math> such that <math>A=B^*B=BB</math>.
== Generalizations ==
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