In this chapter we conclude our analysis of the structure of a linear transformation J : V → V. We derive our deepest structural results, the rational canonical form of J and, when V is a vector space over an algebraically closed field F, the Jordan canonical form of J.

Recall our metaphor of coordinates as giving a language in which to describe linear transformations. A basis B of V in which [J]B is in canonical form is a “right” language to describe the linear transformation J. This is especially true for the Jordan canonical form, which is intimately related to eigenvalues, eigenvectors, and generalized eigenvectors.

The importance of the Jordan canonical form of J cannot be overemphasized. Every structural fact about a linear transformation is encoded in its Jordan canonical form.

We not only show the existence of the Jordan canonical form, but also derive an algorithm for finding the Jordan canonical form of J as well as finding a Jordan basis of V, assuming we can factor the characteristic polynomial CJ(x). (Of course, there is no algorithm for factoring polynomials, as we know from Galois theory.)

We have arranged our exposition in what we think is the clearest way, getting to the simplest (but still important) results as quickly as possible in the preceding chapter, and saving the deepest results for this chapter. However, this is not the logically most economical way.