Trigonalizable/Diagonalizable/Eigen- and generalized eigenspace/Exercise

Let

$\varphi \colon V\longrightarrow V$

be a trigonalizable mapping. Show that ${}\varphi$ is diagonalizable if and only if for every ${}\lambda \in K$, the equality

${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,$

holds.

Create a solution