Eigenvectors of multiplied matrices?

Question

I have the review question if the vector u is an eigenvector of A and and eigenvector of B, then is also an eigenvector of AB, and BA, true or false, and explain why? I just have a feeling its true, but don't know how I would prove this or answer this with any certainty?

Answer 1

Let's say $Au=\lambda u$ and $Bu= \mu u$.

Then $ABu=A(\mu u)=\lambda \mu u$. So, $u$ is an eigenvector of $AB$.

Similar argument applies for $BA$.