# when do $$A$$ and $$AA^T$$ have the same spectrum?

There are matrices for which the spectrum of $$\mat{A}$$ and $$\mat{A}\mat{A}^T$$ are identical. As an example, take
$\mat{A} = \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$
Freaky, right? I wouldn’t have believed it. I know one class of matrices for which this is true, but it’s a bit funky: if $$\mat{C}$$ and $$\mat{B}$$ are rectangular matrices related in a nice way (essentially, $$\mat{B}$$ is $$\mat{C}$$ plus a perturbation that is orthogonal to the range space of $$\mat{C}$$), then $$\mat{A} = \mat{C} \mat{B}^\dagger$$ has this property. Here $$\dagger$$ denotes pseudoinversion. The proof of this special case is complicated, and not particularly intuitive. So I wonder if there’s a simpler proof for the general case.