= 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 $\begin{pmatrix}a&1\\0&a\end{pmatrix}$. Choose an eigenvector and complete it to a basis. The matrix becomes $\begin{pmatrix}a&c\\0&a\end{pmatrix}$ 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.
Back to article page
