if $AB=BA$ , does that mean $A$ and $B$ have same Eigenvectors?

Question

I already know that if $AB = BA$ and $A$ has no repeated eigenvalues, then $A$ and $B$ have the same eigenvectors.

I also know that if $A$ and $B$ have same eigenvectors then, they commute $AB=BA$, and to give an affirmative answer to the question, I need to prove the opposite now with a condition that $A$ has no repeated eigenvalues.

Answer 1

Not true. Take any $n \times n$ matrix $A$ with no repeated eigenvalues, and $B = I$ (the $n \times n$ identity matrix). Then every nonzero vector is an eigenvector of $B$, but most are not eigenvectors of $A$.

What is true is that every eigenvector of $A$ is an eigenvector of $B$.