TITLE: Spectral radius of a matrix product
QUESTION [2 upvotes]: Let $A, B \in \mathbb{R}^{n\times n}$ be two invertible matrices. I want to know if one can always find real scalars $\lambda_1, \ldots,\lambda_p$ such that 
$$\rho\left(\prod_{i=1}^{p} (A- \lambda_i B) \right) < 1$$
where $\rho$ denotes the spectral radius.

REPLY [0 votes]: Take 
$$
A = \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix},\quad
B = \begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}.
$$
They are simoultaneously diagonalizable and the eigenvalues of $A+\lambda B$ are $1\pm\lambda \text i$, that have both norm at least 1. Notice that the eigenvalues of
$\prod_{i=1}^{p} (A- \lambda_i B) $ are 
$\prod_{i=1}^{p} (1- \lambda_i \text i) $ and 
$\prod_{i=1}^{p} (1+ \lambda_i \text i) $, that have both norm at least 1.
{"set_name": "stack_exchange", "score": 2, "question_id": 3648013}