 Repeated-eigenvalue classification in dimension two

A complex two-dimensional linear operator whose two eigenvalues both equal $a$ is similar either to $aI$ or to $\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)$. Choose an eigenvector and complete it to a basis. The matrix becomes $\left(\begin{array}{cc}a& c\\ 0& a\end{array}\right)$ because its characteristic polynomial is $(t-a{)}^2$. A nonzero $c$ can be normalized to one by rescaling the first basis vector. The two types are distinguished by eigenspace dimension.