Cholesky Decomposition

Let's say $A$ is symmetric positive definite matrix, then

$$A = LL^T$$

Where $L$is a lower triangular matrix.

Note the cholesky decomposition is unique i.e matrix $L$is unique for a given $A$.

How to use Cholesky Decomposition

The Cholesky decomposition is mainly used for the numerical solution of linear equations . If A is symmetric and positive definite, then we can solve by first computing the Cholesky decomposition , then solving for y by forward substitution, and finally solving for x by back substitution.

Square Root

For $A = LL^T$, $L$could be thought of as square root of matrix $A$, though mind that it's not a unique square root, many other matrix can be also be thought of as square root of $A$.

Application of Cholesky decomposition

$$\Sigma = LL^T$$

Geometrically, the Cholesky matrix transforms uncorrelated variables into variables whose variances and covariances are given by Σ. In particular, if you generate p standard normal variates, the Cholesky transformation maps the variables into variables for the multivariate normal distribution with covariance matrix Σ and centered at the origin.

Use the Cholesky transformation to correlate and uncorrelate variables - The DO LoopThe DO Loop