Here's what I was thinking:
Let $A$ be a Hermitian matrix whose columns form an orthonormal basis for the finite dimensional vector space. $A$ is then diagonalizable; i.e. $A$ is similar to $\operatorname{diag}\{\lambda_1,\ldots\lambda_n\}$. These $\lambda_i$'s denote the (real) eigenvalues of $A$ as well.
If all these eigenvalues are positive, then we can say $A$ is a positive definite matrix. But how can we determine this? It seems there exist Hermitian matrices whose columns form an orthonormal basis that have negative eigenvalues.
On the other hand, if we assume $A$ is Hermitian and positive definite, can't we always construct an orthonormal basis from $A$ since it's diagonalizable? How does positive definite-ness affect this?