Cholesky decomposition of real symmetric positive semidefinite matrix

Question

Assuming $A$ is a real symmetric positive semidefinite square matrix, the cholesky decomposition applies only when $A$ is strictly positive definite. Otherwise the algorithm will fail according to documentations of a number of linear algebra packages including Matlab. However, if I perform an eigen value decomposition $A=V\Lambda V^\top=PP^\top$, where $P=V\sqrt{\Lambda}$, then a QR decomposition $P^\top=QR$, I get $A=PP^{\top}=R^{\top}Q^{\top}QR=LL^\top$, where $L=R^\top$ is a lower triangular matrix. It seems I have obtained a cholesky decomposition of a semidefinite $A$ without any problem. I was wondering if there is anything wrong here? If not, how can I directly obtain the $A=LL^\top$ decomposition without these highly inefficient procedures?