$A = LDL^T \Rightarrow $all of the main diagonal entries of $D$ are positive? $A$ is symmetric positive definite and $A = LDL^T$, where $L$ is unit lower triangular and $D$ is diagonal. I want to prove that the main-diagonal entries of $D$ are all positive.
I have tried $\det(A)>0 \Rightarrow \det(LDL^T)>0$. Since $\det(L)=\det(L^T)=1$, $\det(D)$ must be positive. But, that doesn't mean all of the main diagonal entries of D are positive. Should I be using a different property?

Let $e_j$ the $j$-th vector of the canonical basis. Since $L$ is invertible we are allowed to write $(L^t)^{-1}e_j$ and since $A$ is positive definite $$0<((L^{-1})^te_j)^tA(L^{-1})^te_j=e_j^tL^{-1}A(L^{-1})^{t}e_j=e_j^tDe_j=D_{jj}$$
and we are done.
