Librería Portfolio

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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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.

First published in 2014.

Leiba Rodman is professor of mathematics at the College of William & Mary. His books include Matrix Polynomials, Algebraic Riccati Equations, and Indefinite Linear Algebra and Applications.



Table of Contents
FrontMatter, pg. i
Contents, pg. vii
Preface, pg. xi
Chapter One. Introduction, pg. 1
Chapter Two. The algebra of quaternions, pg. 9
Chapter Three. Vector spaces and matrices: Basic theory, pg. 28
Chapter Four. Symmetric matrices and congruence, pg. 64
Chapter Five. Invariant subspaces and Jordan form, pg. 83
Chapter Six. Invariant neutral and semidefinite subspaces, pg. 131
Chapter Seven. Smith form and Kronecker canonical form, pg. 153
Chapter Eight. Pencils of hermitian matrices, pg. 172
Chapter Nine. Skewhermitian and mixed pencils, pg. 194
Chapter Ten. Indefinite inner products: Conjugation, pg. 228
Chapter Eleven. Matrix pencils with symmetries: Nonstandard involution, pg. 261
Chapter Twelve. Mixed matrix pencils: Nonstandard involutions, pg. 279
Chapter Thirteen. Indefinite inner products: Nonstandard involution, pg. 300
Chapter Fourteen. Matrix equations, pg. 328
Chapter Fifteen. Appendix: Real and complex canonical forms, pg. 339
Bibliography, pg. 353
Index, pg. 361