In this chapter we generalize to matrices the theory of matrix normal forms presented in Chapter ?? for matrices. In this theory we ask: What is the simplest form that a matrix can have up to similarity. After first presenting several preliminary results, the theory culminates in the Jordan normal form theorem, Theorem ??.
The first of the matrix normal form results — every matrix with distinct real eigenvalues can be diagonalized — is presented in Section ??. The basic idea is that when a matrix has distinct real eigenvalues, then it has linearly independent eigenvectors. In Section ?? we discuss matrix normal forms when the matrix has distinct eigenvalues some of which are complex. When an matrix has fewer than linearly independent eigenvectors, it must have multiple eigenvalues and generalized eigenvectors. This topic is discussed in Section ??. The Jordan normal form theorem is introduced in Section ?? and describes similarity of matrices when the matrix has fewer than independent eigenvectors. The proof is given in Appendix ??.
We introduced Markov matrices in Section ??. One of the theorems discussed there has a proof that relies on the Jordan normal form theorem, and we prove this theorem in Appendix ??.