# If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of $AB$?

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 $AB$?

Hint: Matrix multiplication is associative: $$AB(v)= A(Bv)$$
• Have you any idea whether associativity is really the crucial property? I'm rather fascinated by translating this from the trivial $\mathbb{R}^{n\times n}$ or $\mathbb{C}^{n\times n}$ cases to the quaternions (where the statement in question seems to hold true) or octonions (no idea). Jul 21, 2013 at 17:20
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 $AB$.
• What if the entries are from a strictly skew field, though? $A(\mu v)=\mu A v$ isn't valid then, yet the statement appears to hold also for quaternion matrices. Jul 20, 2013 at 19:25
Note that $(AB)v=A(Bv)=A(\beta v)= \beta(Av)=(\beta\alpha ) v$ with $\alpha$ and $\beta$ being the relevant eigenvalues.