$A$ 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?