What is the relation between the eigenvectors?

Question

For the first part of part (b) I get the result by using a similarity transformation $ x=Rx^{*}$ but I am not sure how to go about the relation between their eigenvectors.

Any help would be much appreciated!

Answer 1

If $v$ is an eigenvector of $B$, then $Rv$ is an eigenvector of $A$: if $\lambda\in\mathbb R$ is such that $Bv=\lambda v$, then$$Bv=\lambda v\iff R^{-1}ARv=\lambda v\iff A(Rv)=\lambda Rv.$$By a similar argument, if $w$ is an eigenvector of $A$, then $R^{-1}w$ is an eigenvector of $B$.

Answer 2

Relation between the eigenvectors of $A$ and $B$:

if $Ax= \mu x$, then put $y=R^{-1}x$ and show that $By= \mu y$.