Prove that every positive semidefinite matrix has nonnegative eigenvalues

Recall the definition of an eigenvalue $\lambda$ (and an eigenvector $\vec{v}$):

$$A\vec{v}=\lambda\vec{v}$$

For a matrix to be positive semi-definite, $\vec{x}^TA\vec{x}\ge0$ for all $\vec{x}$. But if $\vec{v}$ is an eigenvector of $A$, then

$$\vec{v}^T A \vec{v} = \vec{v}^T (\lambda \vec{v}) = \vec{v}^T \vec{v} \lambda$$

Since $\vec{v}^T \vec{v}$ is necessarily a positive number, in order for $\vec{v}^TA\vec{v}$ to be greater than or equal to $0$, $\lambda$ must be greater than or equal to $0$.

Hint: Start with the definition. The $n\times n$ symmetric matrix $A$ is positive semidefinite if $x^TAx\geq 0$ for all $x\in\mathbb{R}^n$.

See

http://en.wikipedia.org/wiki/Positive-definite_matrix#Characterizations.

The first characterization (modified a bit for the semidefinite case) is what you want.