Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition

Theorem
Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$ such that:


 * there exists an invertible matrix $C$ such that $A = C^\intercal C$.

Then $A$ is positive definite.

Proof
Let $A$ be a symmetric matrix such that:


 * there exists an invertible matrix $C$ such that $A = C^\intercal C$.

Let $\mathbf v$ be a non-zero vector.

Then:

So $A$ is positive definite.