Eigenvalues of a composition of matrices I have a quite more general problem that I have reduced to the following question, which I don have any idea how to attack.
Let $A$ and $B$ be square matrices s.t. $AB$ has eigenvalue 1, take $P$ an orthogonal matrix, is 1 still an eigenvalue for $APBP^{-1}$. Taking $B$ a diagonal matrix this is trivially true, so it is not far-fetched to think this is possible, if not I would like a counterexample if possible. Thank in advance.

This is certainly not true already for $2\times 2$-matrices. Here is an example with non-diagonal matrices.
$$
A=\begin{pmatrix} 1 & 2 \cr 3 & 7 \end{pmatrix}, \; 
B=\begin{pmatrix} 7 & -2 \cr -3 & 1 \end{pmatrix}, \; 
P=\frac{1}{3}\cdot \begin{pmatrix} 1 & \sqrt{8} \cr \sqrt{8} & -1 \end{pmatrix}, \; 
$$
Then $AB=I$ has both eigenvalues $1$, but
$$
APBP^{-1}=\frac{1}{9}\begin{pmatrix} 14\sqrt{2}-11 & 4(8\sqrt{2} + 23) \cr 2(27\sqrt{2} - 23) & 106\sqrt{2} + 333 \end{pmatrix}
$$
has no eigenvalue $1$.
