Proving a matrix is positive definite using Cholesky decomposition If you have a Hermitian matrix $C$ that you can rewrite using Cholesky decomposition, how can you use this to show that $C$ is also positive definite?
$C$ is positive definite if $x^\top C x > 0$ and $x$ is a vector.

From Wikipedia:

If A can be written as LL* for some invertible L, lower triangular or
  otherwise, then A is Hermitian and positive definite.

$A=LL^*\implies x^*Ax=(L^*x)^*(L^*x)\ge 0$
Since $L$ is invertible, $L^*x\ne 0$ unless $x=0$, so $x^*Ax>0\ \forall\ x\ne 0$ 
