Editing Cholesky decomposition (section)

=== 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>