TITLE: Classifying complex $2\times 2$ matrices up to similarity
QUESTION [8 upvotes]: I would like to prove the following proposition, which is given as an exercise in Hoffman and Kunze:

If $A$ is a $2\times 2$ matrix with coefficients in $\mathbb{C}$, then $A$ is similar either to a matrix of the form $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ or to a matrix of the form $\begin{pmatrix} a & 0 \\ 1 & a \end{pmatrix}$.

A hint directs the reader to prove that if $N$ is an nilpotent matrix (also in $M_{2}(\mathbb{C})$), then either $N=0$ or $N$ is similar to $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$. I have proven this claim (by supposing that $N \neq 0$ and showing that the transformation induced by multiplication by $N$ has matrix representation $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ under a particular basis).
However, I'm not really sure how to use this knowledge to prove the proposition in question. Any steps in the right direction would be appreciated. Also, I'd love to see any other proofs of the proposition.
Thanks in advance!

REPLY [10 votes]: Hint:
If a matrix has two distinct eigenvalues, it is similar to $\begin{pmatrix} a&0 \\ 0&b\end{pmatrix}$ with $a\neq b$.  Now, suppose that $A$ has the eigenvalue $a$ of algebraic multiplicity $2$.  We note that $A - aI$ must be nilpotent.  From there, note that for any invertible $S$, 
$$
S(A - aI)S^{-1} = SAS^{-1}-SaIS^{-1} = SAS^{-1}-aI
$$