# Definition of Cholesky decomposition

Put content here**Definition:** The **Cholesky decomposition** of a symmetric (Hermitian) positive-definite matrix $A$ is a factorization:
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$$A = LL^*$$
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where $L$ is a lower triangular matrix with positive diagonal entries.
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For real symmetric positive-definite matrices: $A = LL^T$.
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**Example:**
$$\begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$$
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**Properties:**
- Exists and is unique for every symmetric positive-definite matrix
- Computationally about twice as fast as LU decomposition
- Used in Monte Carlo simulation, Kalman filters, and optimization
- Can be used to solve $Ax = b$ via forward and backward substitution
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**Algorithm:** Entries of $L$ are computed column by column:
$l_{jj} = \sqrt{a_{jj} - \sum_{k=1}^{j-1} l_{jk}^2}$ and $l_{ij} = (a_{ij} - \sum_{k=1}^{j-1} l_{ik}l_{jk})/l_{jj}$ for $i > j$.

# Parents

* Factorization of matrices