Editing Cholesky decomposition (section)

=== LDL decomposition ===
An alternative form, eliminating the need to take square roots when {{math|'''A'''}} is symmetric, is the symmetric indefinite factorization<ref>{{cite book |first=D. |last=Watkins |year=1991 |title=Fundamentals of Matrix Computations |url=https://archive.org/details/fundamentalsofma0000watk |url-access=registration |location=New York |publisher=Wiley |page=[https://archive.org/details/fundamentalsofma0000watk/page/84 84] |isbn=0-471-61414-9 }}</ref>
<math display=block>
\begin{align}
\mathbf{A} = \mathbf{LDL}^\mathrm{T} & =
\begin{pmatrix}   1 & 0 & 0 \\
   L_{21} & 1 & 0 \\
   L_{31} & L_{32} & 1\\
\end{pmatrix}
\begin{pmatrix}   D_1 & 0 & 0 \\
   0 & D_2 & 0 \\
   0 & 0 & D_3\\
\end{pmatrix}
\begin{pmatrix}   1 & L_{21} & L_{31} \\
   0 & 1 & L_{32} \\
   0 & 0 & 1\\
\end{pmatrix} \\[8pt]
& = \begin{pmatrix}   D_1 &   &(\mathrm{symmetric})   \\
   L_{21}D_1 & L_{21}^2D_1 + D_2& \\
   L_{31}D_1 & L_{31}L_{21}D_{1}+L_{32}D_2 & L_{31}^2D_1 + L_{32}^2D_2+D_3.
\end{pmatrix}.
\end{align}
</math>

The following recursive relations apply for the entries of {{math|'''D'''}} and {{math|'''L'''}}:
<math display=block> D_j = A_{jj} - \sum_{k=1}^{j-1} L_{jk}^2 D_k, </math>
<math display=block> L_{ij} = \frac{1}{D_j} \left( A_{ij} - \sum_{k=1}^{j-1} L_{ik} L_{jk} D_k \right) \quad \text{for } i>j. </math>

This works as long as the generated diagonal elements in {{math|'''D'''}} stay non-zero. The decomposition is then unique. {{math|'''D'''}} and {{math|'''L'''}} are real if {{math|'''A'''}} is real.

For complex Hermitian matrix {{math|'''A'''}}, the following formula applies:

<math display=block> D_{j} = A_{jj} - \sum_{k=1}^{j-1} L_{jk}L_{jk}^* D_k, </math>
<math display=block> L_{ij} = \frac{1}{D_j} \left( A_{ij} - \sum_{k=1}^{j-1} L_{ik} L_{jk}^* D_k \right) \quad \text{for } i>j. </math>

Again, the pattern of access allows the entire computation to be performed in-place if desired.