Statement
If A has real entries and is symmetric (or more generally, has complex-valued entries and is Hermitian) and positive definite, then A can be decomposed as
- A = LL*,
where L is a lower triangular matrix with strictly positive diagonal entries, and L* denotes the conjugate transpose of L. This is the Cholesky decomposition.
The Cholesky decomposition is unique: given a Hermitian, positive-definite matrix A, there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.
The requirement that L have strictly positive diagonal entries can be dropped to extend the factorization to the positive-semidefinite case. The statement then reads: a square matrix A has a Cholesky decomposition if and only if A is Hermitian and positive semi-definite. Cholesky factorizations for positive-semidefinite matrices are not unique in general.
In the special case that A is a symmetric positive-definite matrix with real entries, L has real entries as well.
Read more about this topic: Cholesky Decomposition
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