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Topics in Quaternion Linear Algebra

Author Leiba Rodman
Release Date: 2014/08/01
ISBN: 9781400852741
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Book Description

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.

Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. 1 Introduction
    1. 1.1 Notation and conventions
    2. 1.2 Standard matrices
  8. 2 The algebra of quaternions
    1. 2.1 Basic definitions and properties
    2. 2.2 Real linear transformations and equations
    3. 2.3 The Sylvester equation
    4. 2.4 Automorphisms and involutions
    5. 2.5 Quadratic maps
    6. 2.6 Real and complex matrix representations
    7. 2.7 Exercises
    8. 2.8 Notes
  9. 3 Vector spaces and matrices: Basic theory
    1. 3.1 Finite dimensional quaternion vector spaces
    2. 3.2 Matrix algebra
    3. 3.3 Real matrix representation of quaternions
    4. 3.4 Complex matrix representation of quaternions
    5. 3.5 Numerical ranges with respect to conjugation
    6. 3.6 Matrix decompositions: nonstandard involutions
    7. 3.7 Numerical ranges with respect to nonstandard involutions
    8. 3.8 Proof of Theorem 3.7.5
    9. 3.9 The metric space of subspaces
    10. 3.10 Appendix: Multivariable real analysis
    11. 3.11 Exercises
    12. 3.12 Notes
  10. 4 Symmetric matrices and congruence
    1. 4.1 Canonical forms under congruence
    2. 4.2 Neutral and semidefinite subspaces
    3. 4.3 Proof of Theorem 4.2.6
    4. 4.4 Proof of Theorem 4.2.7
    5. 4.5 Representation of semidefinite subspaces
    6. 4.6 Exercises
    7. 4.7 Notes
  11. 5 Invariant subspaces and Jordan form
    1. 5.1 Root subspaces
    2. 5.2 Root subspaces and matrix representations
    3. 5.3 Eigenvalues and eigenvectors
    4. 5.4 Some properties of Jordan blocks
    5. 5.5 Jordan form
    6. 5.6 Proof of Theorem 5.5.3
    7. 5.7 Jordan forms of matrix representations
    8. 5.8 Comparison with real and complex similarity
    9. 5.9 Determinants
    10. 5.10 Determinants based on real matrix representations
    11. 5.11 Linear matrix equations
    12. 5.12 Companion matrices and polynomial equations
    13. 5.13 Eigenvalues of hermitian matrices
    14. 5.14 Differential and difference equations
    15. 5.15 Appendix: Continuous roots of polynomials
    16. 5.16 Exercises
    17. 5.17 Notes
  12. 6 Invariant neutral and semidefinite subspaces
    1. 6.1 Structured matrices and invariant neutral subspaces
    2. 6.2 Invariant semidefinite subspaces respecting conjugation
    3. 6.3 Proof of Theorem 6.2.6
    4. 6.4 Unitary, dissipative, and expansive matrices
    5. 6.5 Invariant semidefinite subspaces: Nonstandard involution
    6. 6.6 Appendix: Convex sets
    7. 6.7 Exercises
    8. 6.8 Notes
  13. 7 Smith form and Kronecker canonical form
    1. 7.1 Matrix polynomials with quaternion coefficients
    2. 7.2 Nonuniqueness of the Smith form
    3. 7.3 Statement of the Kronecker form
    4. 7.4 Proof of Theorem 7.3.2: Existence
    5. 7.5 Proof of Theorem 7.3.2: Uniqueness
    6. 7.6 Comparison with real and complex strict equivalence
    7. 7.7 Exercises
    8. 7.8 Notes
  14. 8 Pencils of hermitian matrices
    1. 8.1 Canonical forms
    2. 8.2 Proof of Theorem 8.1.2
    3. 8.3 Positive semidefinite linear combinations
    4. 8.4 Proof of Theorem 8.3.3
    5. 8.5 Comparison with real and complex congruence
    6. 8.6 Expansive and plus-matrices: Singular H
    7. 8.7 Exercises
    8. 8.8 Notes
  15. 9 Skewhermitian and mixed pencils
    1. 9.1 Canonical forms for skewhermitian matrix pencils
    2. 9.2 Comparison with real and complex skewhermitian pencils
    3. 9.3 Canonical forms for mixed pencils: Strict equivalence
    4. 9.4 Canonical forms for mixed pencils: Congruence
    5. 9.5 Proof of Theorem 9.4.1: Existence
    6. 9.6 Proof of Theorem 9.4.1: Uniqueness
    7. 9.7 Comparison with real and complex pencils: Strict equivalence
    8. 9.8 Comparison with complex pencils: Congruence
    9. 9.9 Proofs of Theorems 9.7.2 and 9.8.1
    10. 9.10 Canonical forms for matrices under congruence
    11. 9.11 Exercises
    12. 9.12 Notes
  16. 10 Indefinite inner products: Conjugation
    1. 10.1 H-hermitian and H-skewhermitian matrices
    2. 10.2 Invariant semidefinite subspaces
    3. 10.3 Invariant Lagrangian subspaces I
    4. 10.4 Differential equations I
    5. 10.5 Hamiltonian, skew-Hamiltonian matrices: Canonical forms
    6. 10.6 Invariant Lagrangian subspaces II
    7. 10.7 Extension of subspaces
    8. 10.8 Proofs of Theorems 10.7.2 and 10.7.5
    9. 10.9 Differential equations II
    10. 10.10 Exercises
    11. 10.11 Notes
  17. 11 Matrix pencils with symmetries: Nonstandard involution
    1. 11.1 Canonical forms for ϕ-hermitian pencils
    2. 11.2 Canonical forms for ϕ-skewhermitian pencils
    3. 11.3 Proof of Theorem 11.2.2
    4. 11.4 Numerical ranges and cones
    5. 11.5 Exercises
    6. 11.6 Notes
  18. 12 Mixed matrix pencils: Nonstandard involutions
    1. 12.1 Canonical forms for ϕ-mixed pencils: Strict equivalence
    2. 12.2 Proof of Theorem 12.1.2
    3. 12.3 Canonical forms of ϕ-mixed pencils: Congruence
    4. 12.4 Proof of Theorem 12.3.1
    5. 12.5 Strict equivalence versus ϕ-congruence
    6. 12.6 Canonical forms of matrices under ϕ-congruence
    7. 12.7 Comparison with real and complex matrices
    8. 12.8 Proof of Theorem 12.7.4
    9. 12.9 Exercises
    10. 12.10 Notes
  19. 13 Indefinite inner products: Nonstandard involution
    1. 13.1 Canonical forms: Symmetric inner products
    2. 13.2 Canonical forms: Skewsymmetric inner products
    3. 13.3 Extension of invariant semidefinite subspaces
    4. 13.4 Proofs of Theorems 13.3.3 and 13.3.4
    5. 13.5 Invariant Lagrangian subspaces
    6. 13.6 Boundedness of solutions of differential equations
    7. 13.7 Exercises
    8. 13.8 Notes
  20. 14 Matrix equations
    1. 14.1 Polynomial equations
    2. 14.2 Bilateral quadratic equations
    3. 14.3 Algebraic Riccati equations
    4. 14.4 Exercises
    5. 14.5 Notes
  21. 15 Appendix: Real and complex canonical forms
    1. 15.1 Jordan and Kronecker canonical forms
    2. 15.2 Real matrix pencils with symmetries
    3. 15.3 Complex matrix pencils with symmetries
  22. Bibliography
  23. Index
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