We know that  a positive definite matrix has a  Cholesky decomposition,but I want to know how a Cholesky decomposition can be done for positive semi-definite matrices?The following sentences come from a paper.
"There are two assumptions on the specified correlation matrix R. The first is a general assumption that R is a possible correlation matrix, i.e. that it is a symmetric positive semidefinite matrix with 1’s on the main diagonal. While implementing the algorithm there is no need to check positive semi-definiteness directly, as we do a Cholesky decomposition of the matrix R at the very start. If R is not positive semi-definite, the Cholesky decomposition will fail."
Thank you for your answer.