Book Description
Quaternions are a number system that has become increasingly useful for representing the rotations of objects in threedimensional 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 selfcontained 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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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
 Cover Page
 Title Page
 Copyright Page
 Dedication Page
 Contents
 Preface
 1 Introduction
 2 The algebra of quaternions

3 Vector spaces and matrices: Basic theory
 3.1 Finite dimensional quaternion vector spaces
 3.2 Matrix algebra
 3.3 Real matrix representation of quaternions
 3.4 Complex matrix representation of quaternions
 3.5 Numerical ranges with respect to conjugation
 3.6 Matrix decompositions: nonstandard involutions
 3.7 Numerical ranges with respect to nonstandard involutions
 3.8 Proof of Theorem 3.7.5
 3.9 The metric space of subspaces
 3.10 Appendix: Multivariable real analysis
 3.11 Exercises
 3.12 Notes
 4 Symmetric matrices and congruence

5 Invariant subspaces and Jordan form
 5.1 Root subspaces
 5.2 Root subspaces and matrix representations
 5.3 Eigenvalues and eigenvectors
 5.4 Some properties of Jordan blocks
 5.5 Jordan form
 5.6 Proof of Theorem 5.5.3
 5.7 Jordan forms of matrix representations
 5.8 Comparison with real and complex similarity
 5.9 Determinants
 5.10 Determinants based on real matrix representations
 5.11 Linear matrix equations
 5.12 Companion matrices and polynomial equations
 5.13 Eigenvalues of hermitian matrices
 5.14 Differential and difference equations
 5.15 Appendix: Continuous roots of polynomials
 5.16 Exercises
 5.17 Notes

6 Invariant neutral and semidefinite subspaces
 6.1 Structured matrices and invariant neutral subspaces
 6.2 Invariant semidefinite subspaces respecting conjugation
 6.3 Proof of Theorem 6.2.6
 6.4 Unitary, dissipative, and expansive matrices
 6.5 Invariant semidefinite subspaces: Nonstandard involution
 6.6 Appendix: Convex sets
 6.7 Exercises
 6.8 Notes
 7 Smith form and Kronecker canonical form
 8 Pencils of hermitian matrices

9 Skewhermitian and mixed pencils
 9.1 Canonical forms for skewhermitian matrix pencils
 9.2 Comparison with real and complex skewhermitian pencils
 9.3 Canonical forms for mixed pencils: Strict equivalence
 9.4 Canonical forms for mixed pencils: Congruence
 9.5 Proof of Theorem 9.4.1: Existence
 9.6 Proof of Theorem 9.4.1: Uniqueness
 9.7 Comparison with real and complex pencils: Strict equivalence
 9.8 Comparison with complex pencils: Congruence
 9.9 Proofs of Theorems 9.7.2 and 9.8.1
 9.10 Canonical forms for matrices under congruence
 9.11 Exercises
 9.12 Notes

10 Indefinite inner products: Conjugation
 10.1 Hhermitian and Hskewhermitian matrices
 10.2 Invariant semidefinite subspaces
 10.3 Invariant Lagrangian subspaces I
 10.4 Differential equations I
 10.5 Hamiltonian, skewHamiltonian matrices: Canonical forms
 10.6 Invariant Lagrangian subspaces II
 10.7 Extension of subspaces
 10.8 Proofs of Theorems 10.7.2 and 10.7.5
 10.9 Differential equations II
 10.10 Exercises
 10.11 Notes
 11 Matrix pencils with symmetries: Nonstandard involution

12 Mixed matrix pencils: Nonstandard involutions
 12.1 Canonical forms for ϕmixed pencils: Strict equivalence
 12.2 Proof of Theorem 12.1.2
 12.3 Canonical forms of ϕmixed pencils: Congruence
 12.4 Proof of Theorem 12.3.1
 12.5 Strict equivalence versus ϕcongruence
 12.6 Canonical forms of matrices under ϕcongruence
 12.7 Comparison with real and complex matrices
 12.8 Proof of Theorem 12.7.4
 12.9 Exercises
 12.10 Notes

13 Indefinite inner products: Nonstandard involution
 13.1 Canonical forms: Symmetric inner products
 13.2 Canonical forms: Skewsymmetric inner products
 13.3 Extension of invariant semidefinite subspaces
 13.4 Proofs of Theorems 13.3.3 and 13.3.4
 13.5 Invariant Lagrangian subspaces
 13.6 Boundedness of solutions of differential equations
 13.7 Exercises
 13.8 Notes
 14 Matrix equations
 15 Appendix: Real and complex canonical forms
 Bibliography
 Index
Product Information
 Title: Topics in Quaternion Linear Algebra
 Author(s):
 Release date: August 2014
 Publisher(s): Princeton University Press
 ISBN: 9781400852741