Eigenvectors of $\text{diag}(x) - xx^T$

Question

Given a vector $x$ what are the eigenvectors of $\text{diag}(x) - xx^T$?

My observations so far:

1. If we had only the first term, the matrix would already be diagonal.

2. The second term, on the other hand, has a diagonal form with a single nonzero entry corresponding to the direction parallel to $x$

3. This matrix acts on the vector $(1,1,\ldots,1)$ to yield a multiple of $x$.

4. I also see how to invert the matrix using the Woodbury identity.