The definition of diagonalizable matrix and diagonalizable linear map is given here https://en.wikipedia.org/wiki/Diagonalizable_matrix and the poof that every idempotent matrix is diagonalizable is given in **Proof 1** here Idempotent matrix is diagonalizable? , but I do not understand how the definition of diagonalizable linear map is used in **Proof 1**, could anyone clarify this for me please?

Thanks!