Proof that the eigenvalues of a positive definite matrix are positive

Proof

1. Let $A$ be a positive definite Hermitian $n \times n$ matrix, and let $\lambda$ be an eigenvalue of $A$.
2. By definition there is $x \in \C^n$ with $x \ne 0$ and $Ax = \lambda x$.
3. Let $x^*$ be the conjugate transpose of $x$. Then $x^* x = \sum_{i=1}^n \abs{x_i}^2$, which is a real number.
4. Since $x \ne 0$, some $x_i \ne 0$, so $x^* x > 0$.
5. By step 2, $x^* A x = x^* (\lambda x) = \lambda\, x^* x$.
6. Since $A$ is positive definite and $x \ne 0$, the number $x^* A x$ is real and positive.
7. By steps 4 to 6, $\lambda = x^* A x / x^* x$ is a quotient of two positive reals. So $\lambda$ is a positive real number.