The Cholesky decomposition factors a symmetric positive definite matrix $A$ as:

$ A = LL^T $

where $L$ is a lower triangular matrix with positive diagonal entries.

Requirements:

• $A$ must be symmetric: $A = A^T$
• $A$ must be positive definite: $\mathbf{x}^TA\mathbf{x} > 0$ for all $\mathbf{x} \neq \mathbf{0}$

Applications:

• Solving linear systems: $A\mathbf{x} = \mathbf{b}$ becomes $L(L^T\mathbf{x}) = \mathbf{b}$
• Computing determinants: $\det(A) = (\prod L_{ii})^2$
• Generating correlated random variables
• Numerical optimization (Newton's method)