Editing Cholesky decomposition (section)

== Statement ==
The Cholesky decomposition of a [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] {{math|'''A'''}}, is a decomposition of the form

<math display=block>\mathbf{A} = \mathbf{L L}^{*},</math>

where {{math|'''L'''}} is a [[lower triangular matrix]] with real and positive diagonal entries, and {{math|'''L'''}}* denotes the [[conjugate transpose]] of {{math|'''L'''}}. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.<ref>{{harvtxt|Golub|Van Loan|1996|p=143}}, {{harvtxt|Horn|Johnson|1985|p=407}}, {{harvtxt|Trefethen|Bau|1997|p=174}}.</ref>

The converse holds trivially: if {{math|'''A'''}} can be written as {{math|'''LL'''*}} for some invertible {{math|'''L'''}}, lower triangular or otherwise, then {{math|'''A'''}} is Hermitian and positive definite.

When {{math|'''A'''}} is a real matrix (hence symmetric positive-definite), the factorization may be written
<math display=block>\mathbf{A} = \mathbf{L L}^\mathsf{T},</math>
where {{math|'''L'''}} is a real lower triangular matrix with positive diagonal entries.<ref>{{harvtxt|Horn|Johnson|1985|p=407}}.</ref><ref>{{Cite web|url=https://mathoverflow.net/questions/125960/diagonalizing-a-complex-symmetric-matrix|title=matrices - Diagonalizing a Complex Symmetric Matrix|website=MathOverflow|access-date=2020-01-25}}</ref><ref>{{Cite journal|last1=Schabauer|first1=Hannes|last2=Pacher|first2=Christoph|last3=Sunderland|first3=Andrew G.|last4=Gansterer|first4=Wilfried N.|date=2010-05-01|title=Toward a parallel solver for generalized complex symmetric eigenvalue problems|journal=Procedia Computer Science|series=ICCS 2010|language=en|volume=1|issue=1|pages=437–445|doi=10.1016/j.procs.2010.04.047|issn=1877-0509|doi-access=free}}</ref>

=== Positive semidefinite matrices ===
If a Hermitian matrix {{math|'''A'''}} is only positive semidefinite, instead of positive definite, then it still has a decomposition of the form {{math|1='''A''' = '''LL'''*}} where the diagonal entries of {{math|'''L'''}} are allowed to be zero.<ref>{{harvtxt|Golub|Van Loan|1996|p=147}}.</ref>
The decomposition need not be unique, for example:
<math display=block>\begin{bmatrix}0 & 0 \\0 & 1\end{bmatrix} = \mathbf L \mathbf L^*, \quad \quad \mathbf L=\begin{bmatrix}0 & 0\\ \cos \theta & \sin\theta\end{bmatrix},</math>
for any {{mvar|θ}}. However, if the rank of {{math|'''A'''}} is {{mvar|r}}, then there is a unique lower triangular {{math|'''L'''}} with exactly {{mvar|r}} positive diagonal elements and {{math|''n'' − ''r''}} columns containing all zeroes.<ref>
{{Cite book |last=Gentle |first=James E. |date=1998 |title=Numerical Linear Algebra for Applications in Statistics |isbn=978-1-4612-0623-1 |publisher=Springer |language=en |page= 94}}</ref>

Alternatively, the decomposition can be made unique when a pivoting choice is fixed. Formally, if {{math|'''A'''}} is an {{math|''n'' × ''n''}} positive semidefinite matrix of rank {{mvar|r}}, then there is at least one permutation matrix {{math|'''P'''}} such that {{math|'''P A P'''<sup>T</sup>}} has a unique decomposition of the form {{math|1='''P A P'''<sup>T</sup> = '''L L'''<sup>*</sup>}} with
<math display=inline> \mathbf L =  \begin{bmatrix} \mathbf L_1 & 0 \\ \mathbf L_2 & 0\end{bmatrix} </math>,
where {{math|'''L'''<sub>1</sub>}} is an {{math|''r'' × ''r''}} lower triangular matrix with positive diagonal.<ref>{{Cite book |last=Higham |first=Nicholas J. |chapter-url=http://eprints.maths.manchester.ac.uk/1193/ |title=Reliable Numerical Computation |publisher=Oxford University Press |year=1990 |isbn=978-0-19-853564-5 |editor-last=Cox |editor-first=M. G. |location=Oxford, UK |pages=161–185 |language=en |editor-last2=Hammarling |editor-first2=S. J. |chapter=Analysis of the Cholesky Decomposition of a Semi-definite Matrix}}</ref>