It seems, although I cannot prove, that if we have a matrix $A$ with the eigenvalue $a$ that the eigenvalues $\lambda$ of $$(A+A^n)x = \lambda x \implies \lambda = a+a^n$$
The last term follows by $Ax=\lambda x \implies A^nx=\lambda^nx$.
This however, seems not to be true for $(A+B)x=\lambda x$ iff $B\not=A$.
- Is there any theorem to prove this? If not, what's an example where this is not true?
Added own thoughts:
Is the above perhaps the results of the fact that $A^n$ has the same eigenvectors as $A$? Whereas $A$ and $B$ does not need to have the same eigenvectors.