As Wikipedia tells us, a real $n \times n$ symmetric matrix $G = [g_{ij}]$ is positive definite if $v^TGv >0$ for all $0 \neq v \in \mathbb{R}^n$. By a well-known theorem of linear algebra it can be shown that $G$ is positive definite if and only if the eigenvalues of $G$ are positive. Therefore, this gives us two distinct ways to say what it means for a matrix to be positive definite.
In Amann and Escher's Analysis II, exercise 7.1.8 seems to provide yet another way recognize a positive definite matrix. In this exercise, $G$ is defined to be positive definite if there exists a positive number $\gamma$ such that $ \sum\limits_{i,j = 1}^n g_{ij}v^iv^j \geq \gamma |v|^2 $
I have not before seen this characterization of a positive definite matrix and I have not been successful at demonstrating that this characterization is equivalent to the other two characterizations listed above.
Can anyone provide a hint how one might proceed to demonstrate this apparent equivalence or suggest a reference that discusses it?