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Let $A$ and $B$ be $n \times n$ complex matrices that commute: $A B=B A$. If $\lambda$ is an eigenvalue of $A$, let $\mathcal{V}_{\lambda}$ be the subspace of all eigenvectors having this eigenvalue.
a) Show there is an vector $v \in \mathcal{V}_{\lambda}$ that is also an eigenvector of $B$, possibly with a different eigenvalue.
b) Give an example showing that some vectors in $\mathcal{V}_{\lambda}$ may not be an eigenvectors of B.
c) If all the eigenvalues of $A$ are distinct (so each has algebraic multiplicity one), show that there is a basis in which both $A$ and $B$ are diagonal. Also, give an example showing this may be false if some eigenvalue of $A$ has multiplicity greater than one.
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