First, let's recall the definition of the Cholesky decomposition: Given a symmetric positive definite square matrix X, the Cholesky decomposition of X is the factorization X=U'U, where U is the square root matrix of X, and satisfies:

(1) U'U = X
(2) U is upper triangular (that is, it has all zeros below the diagonal).

It seems that the assumption of positive definiteness is necessary. Actually, it is "positive definite" which guarantees the existence of such kind of decomposition. 

平均因子分解法，適用于正定矩陣