**Abstract:** The aim of this note is to compute the Grothendieck group of the category of endomorphisms. This computation mostly plays with linear algebra. The main result is that in , every endomorphism is uniquely characterized by and its characteristic polynomial . This computation was due to [1]. We will explain how to think about this computation, the reason for certain constructions and the “diagonalization” in this computation.

Edit history: fixed some mislabeling of diagrams (Mar 21 2022)

[1] Almkvist, G. (1974). The Grothendieck ring of the category of endomorphisms. *Journal of Algebra*, *28*(3), 375-388.