I wish to prove that any $2\times 2$ matrix $T$ with only one eigenvalue $ \lambda$ of geometric multiplicity 1 is similar to one of the form
$\left[\begin{array}{cc} \lambda & 1 \\ 0 &\lambda \end{array}\right]$.
Now if I choose my basis to be $v$, where $v$ is some eigenvector of $\lambda$ and any vector $w$ not in null $(T - \lambda I)$ this should work. It is obvious in this basis how the first column of the matrix above comes about,
and for the second column, we note that
$T(w) = (T-\lambda I)w + \lambda w$ and by Cayley-Hamilton it is clear that $(T - \lambda I)w$ is an eigenvector of $T$. However this only means that in the matrix above I have a constant, not necessarily 1 in the slot where it is sitting.
How can I choose $w$ so that $(T-\lambda I)w$ is an eigenvector of $T$ of eigenvalue 1?
Thanks.