Show $ \begin{bmatrix} A & BC(A+BC) \\ I_n & 0 \end{bmatrix} $ and $\begin{bmatrix} A+BC & 0 \\ 0 & -CB \end{bmatrix}$ have same nonzero eigenvalues

Question

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times n}$. Define \begin{equation} D_1=\begin{bmatrix} A & BC(A+BC) \\ I_n & 0 \end{bmatrix},\qquad D_2=\begin{bmatrix} A+BC & 0 \\ 0 & -CB \end{bmatrix} \end{equation} Show that $D_1$ and $D_2$ have the same nonzero eigenvalues.

My attemp: If we check the claim numerically, we see that the nonzero eigenvalues of $D_1$ and $D_2$ are the same. Probably, there is a factorization of $D_1$, for example $D_1=EF$ such that $D_2=FE$.

Answer 1

In general, $BC$ and $CB$ share the same (multi)set of nonzero eigenvalues. The conclusion now follows from the observation that your $D_1$ is similar to \begin{aligned} &\pmatrix{I_n&-(A+BC)\\ 0&I_n}\pmatrix{A&BC(A+BC)\\ I_n&0}\pmatrix{I_n&A+BC\\ 0&I_n}\\ &=\pmatrix{-BC&BC(A+BC)\\ I_n&0}\pmatrix{I_n&A+BC\\ 0&I_n}\\ &=\pmatrix{-BC&0\\ I_n&A+BC}. \end{aligned}