Cyclic Subspaces, Duality and the Jordan Canonical Form

Abstract

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 f. Because of its great importance there have been many different derivations of this form using different mathematical tools.