Jordan Canonical Form
Subtitle: theory and practice
By: Steven H. Weintraub
Series: Synthesis lectures on mathematics and statistics -- # 6
Publisher: Morgan & Claypool Publishers
Published: 2009
Place of publication: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA)
Language: eng
Format: [electronic resource] :
Number of pages: 96
ISBN-13: 9781608452514; 9781608452507
Subjects: Linear Algebras; Jordan algebras; Eigenvalues
Other titles: Synthesis digital library of engineering and computer science.
Description: 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 labelled eigenstructure picture (ESP) of A, determines P as well.We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.
Notes: Part of: Synthesis digital library of engineering and computer science.

Title from PDF t.p. (viewed on September 9, 2009).

Series from website.

Includes index.

Abstract freely available; full-text restricted to subscribers or individual document purchasers.

Also available in print.

Mode of access: World Wide Web.

System requirements: Adobe Acrobat reader.
Table of contents: title: 1. Fundamentals on vector spaces and linear transformations; title: Bases and coordinates; title: Linear transformations and matrices; title: Some special matrices; title: Polynomials in T and A; title: Subspaces, complements, and invariant subspaces; title: 2. The structure of a linear transformation; title: Eigenvalues, eigenvectors, and generalized eigenvectors; title: The minimum polynomial; title: Reduction to BDBUTCD form; title: The diagonalizable case; title: Reduction to Jordan Canonical Form; title: Exercises; title: 3. An algorithm for Jordan Canonical Form and Jordan Basis; title: The ESP of a linear transformation; title: The algorithm for Jordan Canonical Form; title: The algorithm for a Jordan Basis; title: Examples; title: Exercises; title: A. Answers to odd-numbered exercises; title: Notation; title: Index.