Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Cholesky decomposition

Builds on Positive definite matrix Lower triangular matrix

Statement

Let $A$ be a positive definite $n \times n$ complex Hermitian matrix. Then there is exactly one lower triangular matrix $L$ with positive real diagonal entries such that $A = L L^*.$ If $A$ is real, then $L$ is real.

Remarks

The factor $L$ is the Cholesky factor of $A$ . For example, $\begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ . It needs about half the work of an LU decomposition, and it solves systems $Ax = b$ with positive definite $A$ . A positive semidefinite matrix also has a decomposition $A = L L^*$ , but $L$ need not be unique.

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Matrix definition
Square matrix definition
Transpose definition
Lower triangular matrix definition
Symmetric matrix definition
Conjugate transpose definition
Hermitian matrix definition
Positive definite matrix definition
Cholesky decomposition statement

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Tag	`0WKG8CYY6A`
URI	`m2://0WKG8CYY6A`
Kind	statement, theorem
Writing	W1 stated
MSC	15A23, 65F05
Status	This copy of the site has no build data. Run `m2 status 0WKG8CYY6A`.
Uses	Positive definite matrix, Lower triangular matrix
Same as	Wikidata Q515375 Wikipedia Cholesky decomposition

Cholesky decomposition

Statement

Remarks

Reading path

Where this leads

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