As we learned in Chapter 5, the advantage of a diagonalizable linear operator lies in the simplicity of its description. Such an operator has a diagonal matrix representation, or, equivalently, there is an ordered basis for the underlying vector space consisting of eigenvectors of the operator. However, not every linear operator is diagonalizable, even if its characteristic polynomial splits. Example 3 of Section 5.2 describes such an operator.
It is the purpose of this chapter to consider alternative matrix representations for nondiagonalizable operators. These representations are called canonical forms. There are different kinds of canonical forms, and their advantages and disadvantages depend on how they are applied. Every canonical form of an operator is obtained by an appropriate choice of an ordered basis. Naturally, the canonical forms of a linear operator are not diagonal matrices if the linear operator is not diagonalizable.
In this chapter, we treat two common canonical forms. The first of these, the Jordan canonical form, requires that the characteristic polynomial of the operator splits. This form is always available if the underlying field is algebraically closed, that is, if every polynomial with coefficients from the field splits. For example, the field of complex numbers is algebraically closed by the fundamental theorem of algebra (see Appendix D). The first two sections deal with this form. The rational canonical form, treated in Section 7.4, does not require such a factorization.