Prove that $LL^T$ is PD

Question

Given a matrix L which is a lower triangular matrix with n raws and n columns, how may I prove that the following matrix is Positive Definite:

$$LL^T$$

In other words, I need to prove that for each $x!=0$, $$x^TLL^Tx>0$$

Answer 1

This is true only for non-singular matrices. $x^{T}LL^{T}x=\|Lx\|^{2}$. So the statement $x^{T}LL^{T}x>0$ whenever $x \neq 0$ is true iff $Lx=0$ implies $x=0$ iff $L$ is non-singular.