Positive semi-definiteness of $\sum_i \alpha_i x_ix_i^T$

Question

Assume that we have $N$ column vectors $x_i; \ i=1,\ldots, N$, and $N$ real numbers $\alpha_i; \ i=1,\ldots,N$.

Can we write down a necessary condition for $\alpha_i$'s under which $\sum_{i=1}^N \alpha_i x_ix_i^T$ is positive semi-definite?

(A sufficient condition would be $\alpha_i\geq 0$, but I want a necessary condition.)