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.

If $C = LL^{\ast}$ with $L$ lower triangular, then
$$ \langle Cx, x \rangle = \langle LL^{\ast} x, x \rangle = \langle L^{\ast} x, L^{\ast} x \rangle \geq 0$$
Thus $C$ is positive semi-definite. Note that, as stated, you cannot show that $C$ is positive definite unless you also know that $L$ is invertible.
