Topics in quaternion linear algebra /
Leiba Rodman.
- 1 online resource
- Princeton series in applied mathematics .
- Princeton series in applied mathematics. .
Includes bibliographical references and index.
Introduction -- The algebra of quaternions -- Vector spaces and matrices: basic theory -- Symmetric matrices and congruence -- Invariant subspaces and Jordan form -- Invariant neutral and semidefinite subspaces -- Smith form and Kronecker canonical from -- Pencils of hermitian matrices -- Skewhermitian and mixed pencils -- Indefinite inner products: conjugation -- Matrix pencils with symmetries: nonstandard involution -- Mixed matrix pencils: nonstandard involutions -- Indefinite inner products: nonstandard involution -- Matrix equations -- Appendix: real and complex canonical forms.
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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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.