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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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

Algebras, Linear --- Quaternions --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Linear algebra --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- Cholesky factorization. --- Hamiltonian matrices. --- Jordan canonical form. --- Jordan form. --- Kronecker canonical form. --- Kronecker form. --- Kronecker forms. --- Schur triangularization theorem. --- Smith form. --- Sylvester equation. --- algebraic Riccati equations. --- antiautomorphisms. --- automorphisms. --- bilateral quadratic equations. --- boundedness. --- canonical forms. --- complex hermitian matrices. --- complex matric pencils. --- complex matrices. --- complex matrix polynomials. --- congruence. --- conjugation. --- conventions. --- determinants. --- diagonal form. --- diagonalizability. --- differential equations. --- dissipative matrices. --- eigenvalues. --- eigenvectors. --- equivalence. --- expansive matrices. --- hermitian inner product. --- hermitian matrices. --- hermitian matrix pencils. --- hermitian pencils. --- indefinite inner products. --- inertia theorems. --- invariant Langragian subspaces. --- invariant Langrangian subspaces. --- invariant neutral subspaces. --- invariant semidefinite subspaces. --- invariant subspaces. --- involutions. --- linear quadratic regulators. --- matrix algebra. --- matrix decompositions. --- matrix equations. --- matrix pencils. --- matrix polynomials. --- maximal invariant semidefinite subspaces. --- metric space. --- mixed matrix pencils. --- mixed pencils. --- mixed quaternion matrix pencils. --- neutral subspaces. --- nondegenerate. --- nonstandard involution. --- nonstandard involutions. --- nonuniqueness. --- notations. --- numerical cones. --- numerical ranges. --- pencils. --- polynomial matrix equations. --- quadratic maps. --- quaternion algebra. --- quaternion coefficients. --- quaternion linear algebra. --- quaternion matrices. --- quaternion matrix pencils. --- quaternion subspaces. --- quaternions. --- real linear transformations. --- real matrices. --- real matrix pencils. --- real matrix polynomials. --- real symmetric matrices. --- root subspaces. --- scalar quaternions. --- semidefinite subspaces. --- skew-Hamiltonian matrices. --- skewhermitian inner product. --- skewhermitian matrices. --- skewhermitian pencils. --- skewsymmetric matrices. --- square-size quaternion matrices. --- standard matrices. --- symmetric matrices. --- symmetries. --- symmetry properties. --- unitary matrices. --- vector spaces.