I know that a matrix $A$ is positive definite and symmetric if and only if there exists a lower triangular matrix $L$ with nonzero diagonal such that $A = LL^T$.
I'm wondering if it similarly holds that a matrix $A$ is positive semidefinite and symmetric if and only if there exists a lower triangular matrix L (where zeros are permitted on the diagonal) such that $A=LL^T$.
The if part is easy since $x^TAx = x^TLL^Tx=||L^Tx||^2 \geq0$. The equality holds when $L^Tx=0$ which can only occur if there are zeros on the diagonals of $L$. Is the converse true? I ask because I want to know if I can use cholesky factorization to check if a hessian is PSD.