Cholesky Decomposition

Given a symmetric positive definite matrix A, the Cholesky decomposition is an upper triangular matrix U with strictly positive diagonal entries such that

$$A=LL^{T}$$

• When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.(wiki)

• Symmetric matrix is positive definite iff It has a unique Cholesky decomposition.

• Symmetric matrix is positive definite iff all its eigenvalues are positive.
• The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL'. However, the decomposition need not be unique when A is positive semidefinite