TITLE: If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of $AB$?
QUESTION [5 upvotes]: I'm preparing for my final and this question came up in one of the practices. I am tempted to say no, but I've been having trouble proving this.
If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of $AB$?

REPLY [13 votes]: Hint: Matrix multiplication is associative: $$AB(v)= A(Bv)$$

REPLY [5 votes]: If $v$ is an eigenvector associated to the eigenvalue $\lambda$ of $A$ and $\mu$ of $B$ then
$$ABv=A(Bv)=A(\mu v)=\mu Av=\mu\lambda v$$
so $v$ is an eigenvector associated to the eigenvalue $\mu\lambda$ of $AB$.

REPLY [3 votes]: Note that $(AB)v=A(Bv)=A(\beta v)= \beta(Av)=(\beta\alpha ) v$ with $\alpha$ and $\beta$ being the relevant eigenvalues.