Linear mapping/C/Jordan normal form/Equivalence relation/Exercise

We consider on the sets of all linear mappings ${}\varphi \colon \mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n}$ the relation ${}\sim$ given by ${}\varphi \sim \psi$ if there exist bases ${}v_{1},\ldots ,v_{n}$ and ${}w_{1},\ldots ,w_{n}$ such that ${}M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )=M_{\mathfrak {w}}^{\mathfrak {w}}(\psi )$ holds. Show that this is an equivalence relation. How can we interpret the Theorem about the Jordan normal form in the context of this equivalence relation?

Create a solution