Prove ${\rm spec}\left(\begin{bmatrix}A&B\\CA&CB\end{bmatrix}\right)\subset {\rm OUD}$

Question

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times n}$. Assume that ${\rm spec}(A+BC)$ is in the open-unit disk. Prove or disprove that ${\rm spec}\left(\begin{bmatrix}A&B\\CA&CB\end{bmatrix}\right)$ is in the open-unit disk.

I have verified this numerically. Any help is appreciated.

Answer 1

It's true. It follows from the fact that $\pmatrix{A&B\\ CA&CB}=\pmatrix{I\\ C}\pmatrix{A&B}$ and $\pmatrix{A&B}\pmatrix{I\\ C}=A+BC$ have the same set of nonzero eigenvalues (counting multiplicity).