Jordan normal form of conjugation on two-by-two matrices
= Jordan normal form of conjugation on two-by-two matrices
{c}
If $\beta$ has eigenvalues $\lambda,\mu\ne0$ and is diagonalizable on a two-dimensional complex vector space, then
$$
\operatorname{JNF}(\phi_\beta)
=\operatorname{diag}\left(1,1,\frac\mu\lambda,\frac\lambda\mu\right).
$$
If $\beta$ has one size-two Jordan block, then
$$
\operatorname{JNF}(\phi_\beta)=J_3(1)\oplus J_1(1).
$$