Jordan Canonical Form: Theory and Practice (Synthesis Lectures on Mathematics and Statistics)

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 (lESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader. Table of Contents: Fundamentals on Vector Spaces and Linear Transformations / The Structure of a Linear Transformation / An Algorithm for Jordan Canonical Form and Jordan Basis 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. 1. Fundamentals On Vector Spaces And Linear Transformations -- Bases And Coordinates -- Linear Transformations And Matrices -- Some Special Matrices -- Polynomials In T And A -- Subspaces, Complements, And Invariant Subspaces -- 2. The Structure Of A Linear Transformation -- Eigenvalues, Eigenvectors, And Generalized Eigenvectors -- The Minimum Polynomial -- Reduction To Bdbutcd Form -- The Diagonalizable Case -- Reduction To Jordan Canonical Form -- Exercises -- 3. An Algorithm For Jordan Canonical Form And Jordan Basis -- The Esp Of A Linear Transformation -- The Algorithm For Jordan Canonical Form -- The Algorithm For A Jordan Basis -- Examples -- Exercises -- A. Answers To Odd-numbered Exercises -- Notation. Steven H. Weintraub. Includes Index. Preface......Page 11 Bases and Coordinates......Page 13 Linear Transformations and Matrices......Page 16 Some Special Matrices......Page 20 Polynomials in T and A......Page 23 Subspaces, Complements, and Invariant Subspaces......Page 25 Eigenvalues, Eigenvectors, and Generalized Eigenvectors......Page 29 The Minimum Polynomial......Page 35 Reduction to BDBUTCD Form......Page 38 The Diagonalizable Case......Page 43 Reduction to Jordan Canonical Form......Page 45 Exercises......Page 52 The ESP of a Linear Transformation......Page 55 The Algorithm for Jordan Canonical Form......Page 60 The Algorithm for a Jordan Basis......Page 65 Examples......Page 68 Exercises......Page 87 Answers to Exercises--Chapter 2......Page 95 Answers to Exercises--Chapter 3......Page 100 Notation......Page 105 Index......Page 107