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Matrix Analysis, Second Edition

Book Description

Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of this acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Preface to the Second Edition
  5. Preface to the First Edition
  6. 0  Review and Miscellanea
    1. 0.0    Introduction
    2. 0.1    Vector spaces
    3. 0.2    Matrices
    4. 0.3    Determinants
    5. 0.4    Rank
    6. 0.5    Nonsingularity
    7. 0.6    The Euclidean inner product and norm
    8. 0.7    Partitioned sets and matrices
    9. 0.8    Determinants again
    10. 0.9    Special types of matrices
    11. 0.10  Change of basis
    12. 0.11  Equivalence relations
  7. 1  Eigenvalues, Eigenvectors, and Similarity
    1. 1.0    Introduction
    2. 1.1    The eigenvalue–eigenvector equation
    3. 1.2    The characteristic polynomial and algebraic multiplicity
    4. 1.3    Similarity
    5. 1.4    Left and right eigenvectors and geometric multiplicity
  8. 2  Unitary Similarity and Unitary Equivalence
    1. 2.0    Introduction
    2. 2.1    Unitary matrices and the QR factorization
    3. 2.2    Unitary similarity
    4. 2.3    Unitary and real orthogonal triangularizations
    5. 2.4    Consequences of Schur’s triangularization theorem
    6. 2.5    Normal matrices
    7. 2.6    Unitary equivalence and the singular value decomposition
    8. 2.7    The CS decomposition
  9. 3  Canonical Forms for Similarity and Triangular Factorizations
    1. 3.0    Introduction
    2. 3.1    The Jordan canonical form theorem
    3. 3.2    Consequences of the Jordan canonical form
    4. 3.3    The minimal polynomial and the companion matrix
    5. 3.4    The real Jordan and Weyr canonical forms
    6. 3.5    Triangular factorizations and canonical forms
  10. 4  Hermitian Matrices, Symmetric Matrices, and Congruences
    1. 4.0    Introduction
    2. 4.1    Properties and characterizations of Hermitian matrices
    3. 4.2    Variational characterizations and subspace intersections
    4. 4.3    Eigenvalue inequalities for Hermitian matrices
    5. 4.4    Unitary congruence and complex symmetric matrices
    6. 4.5    Congruences and diagonalizations
    7. 4.6    Consimilarity and condiagonalization
  11. 5  Norms for Vectors and Matrices
    1. 5.0    Introduction
    2. 5.1    Definitions of norms and inner products
    3. 5.2    Examples of norms and inner products
    4. 5.3    Algebraic properties of norms
    5. 5.4    Analytic properties of norms
    6. 5.5    Duality and geometric properties of norms
    7. 5.6    Matrix norms
    8. 5.7    Vector norms on matrices
    9. 5.8    Condition numbers: inverses and linear systems
  12. 6  Location and Perturbation of Eigenvalues
    1. 6.0    Introduction
    2. 6.1    Gersgorin discs
    3. 6.2    Gersgorin discs – a closer look
    4. 6.3    Eigenvalue perturbation theorems
    5. 6.4    Other eigenvalue inclusion sets
  13. 7  Positive Definite and Semidefinite Matrices
    1. 7.0    Introduction
    2. 7.1    Definitions and properties
    3. 7.2    Characterizations and properties
    4. 7.3    The polar and singular value decompositions
    5. 7.4    Consequences of the polar and singular value decompositions
    6. 7.5    The Schur product theorem
    7. 7.6    Simultaneous diagonalizations, products, and convexity
    8. 7.7    The Loewner partial order and block matrices
    9. 7.8    Inequalities involving positive definite matrices
  14. 8  Positive and Nonnegative Matrices
    1. 8.0    Introduction
    2. 8.1    Inequalities and generalities
    3. 8.2    Positive matrices
    4. 8.3    Nonnegative matrices
    5. 8.4    Irreducible nonnegative matrices
    6. 8.5    Primitive matrices
    7. 8.6    A general limit theorem
    8. 8.7    Stochastic and doubly stochastic matrices
  15. Appendix A Complex Numbers
  16. Appendix B Convex Sets and Functions
  17. Appendix C The Fundamental Theorem of Algebra
  18. Appendix D Continuity of Polynomial Zeroes and Matrix Eigenvalues
  19. Appendix E Continuity, Compactness, and Weierstrass’s Theorem
  20. Appendix F Canonical Pairs
  21. References
  22. Notation
  23. Hints for Problems
  24. Index