Editing Cholesky decomposition (section)

== Generalization ==
The Cholesky factorization can be generalized {{Citation needed|date=October 2016}} to (not necessarily finite) matrices with operator entries. Let <math display=inline>\{\mathcal{H}_n \}</math> be a sequence of [[Hilbert spaces]]. Consider the operator matrix

<math display=block>
\mathbf{A} =
\begin{bmatrix}
\mathbf{A}_{11}   & \mathbf{A}_{12}   & \mathbf{A}_{13} & \; \\
\mathbf{A}_{12}^* & \mathbf{A}_{22}   & \mathbf{A}_{23} & \; \\
\mathbf{A} _{13}^* & \mathbf{A}_{23}^* & \mathbf{A}_{33} & \; \\
\;       & \;       & \;     & \ddots
\end{bmatrix}
</math>

acting on the direct sum

<math display=block>\mathcal{H} = \bigoplus_n \mathcal{H}_n,</math>

where each

<math display=block>\mathbf{A}_{ij} : \mathcal{H}_j \rightarrow \mathcal{H} _i</math>

is a [[bounded operator]]. If {{math|'''A'''}} is positive (semidefinite) in the sense that for all finite {{mvar|k}} and for any

<math display=block>h \in \bigoplus_{n = 1}^k \mathcal{H}_k ,</math>

there is <math display=inline>\langle h, \mathbf{A} h\rangle \ge 0</math>, then there exists a lower triangular operator matrix {{math|'''L'''}} such that {{math|1='''A''' = {{math|'''LL'''*}}}}. One can also take the diagonal entries of {{math|'''L'''}} to be positive.