QUESTION:

Let $\displaystyle G$ be an $\displaystyle n \times n$ matrix with a factorization $\displaystyle G = LDL^{T}$, where $\displaystyle L$ is a unit lower triangular matrix and $\displaystyle D$ is a diagonal matrix. Show that $\displaystyle G$ is positive definite if and only if $\displaystyle D_{ii} > 0$, for $\displaystyle i=1,2,\ldots,n$.

ATTEMPT:

Assume $\displaystyle D_{ii}>0$ for $\displaystyle i=1,2,\ldots,n$. Then $\displaystyle D = D^{\frac{1}{2}} D^{\frac{1}{2}}$ is possible and

$\displaystyle x^T G x = x^T L D L^T x = x^T L D^{\frac{1}{2}} D^{\frac{1}{2}} L^T x = x^T (LD^{\frac{1}{2}}) (L D^{\frac{1}{2}})^T x = ( (LD^{\frac{1}{2}})^T x)^T (L D^{\frac{1}{2}})^T x = y^T y = \sum^{n}_{i=1} y_i^2 \geq 0$

where $\displaystyle y = (LD^{\frac{1}{2}})^T x$.

Because $\displaystyle LD^{\frac{1}{2}}$ has $\displaystyle n$ pivots it is nonsingular and hence $\displaystyle y = 0$ if and only if $\displaystyle x = 0$. Therefore $\displaystyle x^T G x > 0$ for all $\displaystyle x \ne 0$.

Assume $\displaystyle x^T G x = x^T LDL^T x$ is positive definite. How can I prove that $\displaystyle D$ has positive diagonals?