Consider you have two $n \times n$ matrices $A,B$ with the same eigenvalue $\pi$. Then $A-B$ has an eigenvalue of $0$.
The question is, is this correct or not?
I was looking for properties in my head and in the course text, but i didn't find anything useful. Cause we don't know if $\pi$ is die only eigenvalue. We also don't know if the eigen vectors corresponding to the eigenvalues are the same. So we don't know if the matrices are similar or not. The only thing I tried was:
Consider that they have the same eigen vector. Then you could write:
$Av-Bv$ = $(A-B)v$ and $\pi v-\pi v = (\pi-\pi) v$ and $\pi-\pi = 0$
So then you could proof this, but this is not the case unfortunately. I can also find no example that it's not true.