Let $V$ be the set of all $n\times n$ matrices over a field $F$. Let $A$ be a fixed element of $V$. Define a linear operator $T$ on $V$ by $T(B)=AB$. I am trying to show that if $\lambda$ is an eigenvalue of $A$, then $\lambda$ is also an eigenvalue of $T$.
So suppose $Av=\lambda v$ for some $v\neq 0$ in $V$ and $\lambda\in F$. So I'd like to prove the existence of a matrix $B$ such that $T(B)=AB=\lambda A$, or equivalently, show that $T-\lambda I_V$ is not invertible (or injective or surjective). But I am not sure how to proceed from here. What can I do?