Full rank Cholesky factorization for rank deficient matrices

Let A be a rank deficient square matrix. We characterize the unique full rank Cholesky factorization LALAT of A where the factor LA is a lower echelon matrix with positive leading entries. We compute an extended decomposition for the normal matrix BTB where B is a rectangular rank deficient matrix. This decomposition is obtained without interchange of rows and without computing all entries of the normal matrix. Algorithms to compute both factorizations are given.