Eigenvalues of block matrix with two matrices

I try to find eigenvalues of this block matrix \begin{equation} M=\left[ \begin{array}{cc} O & I_{n} \\ -AB & -A-B-I_{n} \end{array} \right] \end{equation} where $I_{n}$ is $n\times n$ identity matrix and $A,B$ are $n\times n$ matrices. Eigenvalues of $A,B$ are known.

To find eigenvalues of $M$, I have this equation. \begin{eqnarray} &&det(\lambda I_{2n}-M) \\ =&&det\left\{\lambda^2 I_n+\lambda(A+B+I_n)+AB\right\} = 0 \end{eqnarray}

Is there a solution of above equation? I can use eigenvalues of $A, B$.

If anything is unclear, please let me know.

• If you are in the lucky position that $A$ and $B$ have the same eigen-directions $A=T^{-1} \Lambda^{(1)} T$, $B=T^{-1} \Lambda^{(2)} T$ you can transform the system via $\bar M := \begin{bmatrix}T^{-1}&0&0&T^{-1}\end{bmatrix} M \begin{bmatrix}T&0&0&T\end{bmatrix}$. In this case the system decomposes into $n$ systems of size $2\times 2$ which are easy to solve ($\lambda^2 + (1+\lambda_A+\lambda_B)\lambda + \lambda_A\lambda_B = 0$). Otherwise you are out of luck (AFAIK) since you cannot deduce the eigenvalues of $M$ from that ones of $A$ and $B$. – Tobias Nov 9 '16 at 8:30
• Sorry, the trafo went wrong: $\bar M := \begin{bmatrix}T^{-1}&0\\0&T^{-1}\end{bmatrix} M \begin{bmatrix}T&0\\0&T\end{bmatrix}$. Furthermore, $\lambda_A$ and $\lambda_B$ are eigenvalues on the same row in $\Lambda^{(1)}$ and $\Lambda^{(2)}$. – Tobias Nov 9 '16 at 8:54