Proving if A is an Hermitian matrix with nonnegative eigenvalues, A is positive semidefinite.

Question

I'm trying to show that if A is an Hermitian matrix with non-negative eigenvalues, then A is positive semi-definite.

The only thing I've thought of so far is using Spectral Theorem. I know I want to show that $x^*Ax\ge0, \forall x\in\mathbb{C}^n$, therefore I need to show the following (the ... part):

$x^*Ax = x^*U^*DUx = ... \ge0$, where $U$ is unitary, $D$ is diagonal with non-negative entries.

Thank you for any assistance.

Answer 1

Please let me know if there is an issue with this answer:

Let $D = E^*E$ where E is some diagonal matrix.

$x^*Ax = x^*U^*DUx = x^*U^*E^*EUx = (EUx)^*(EUx) = \left< EUx,EUx\right>\ge0$. The final step is due to properties of inner products of complex vectors.