In this chapter we use the duality theory to analyze the properties of an endomorphism f on a finite dimensional vector space \(\mathcal V\) in detail. We are particularly interested in the algebraic and geometric multiplicities of the eigenvalues of f and the characterization of the corresponding eigenspaces. Our strategy in this analysis is to decompose the vector space \(\mathcal V\) into a direct sum of f-invariant subspaces so that, with appropriately chosen bases, the essential properties of f will be obvious from its matrix representation. The matrix representation that we derive is called the Jordan canonical form of endomorphisms and matrices. It exists whenever the characteristic polynomial of f resp. A decomposes into linear factors over the given field. Because of its great importance, there have been many different derivations of this form using different mathematical tools since its discovery in the nineteenth century. Our approach using duality theory is based on an article by Vlastimil Pták (1925–1999) from 1956. Using the same strategy, in this chapter we also derive the Frobenius canonical form of endomorphisms and matrices. Unlike the Jordan canonical form, the Frobenius canonical form exists even when f resp. A does not have eigenvalues.