Diagonalizable matrix and diagonalizable linear map. The definition of diagonalizable matrix and diagonalizable linear map is given here < 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!

He doesn't use that definition, but he showed that $A:V\longrightarrow V$ is a diagonalizable linear map. Indeed he found a basis such that the associated linear map is represented by a diagonal matrix.