Jordan Canonical Form : Theory and Practice

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra.

The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix.

This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials.

We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem.

Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V ?

V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries.

Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1.

We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A.

In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear.

The ESP of A determines J, and a refinement, the labeled eigenstructure picture (?ESP) of A, determines P as well.

We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.