*Leiba Rodman*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161853
- eISBN:
- 9781400852741
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161853.001.0001
- Subject:
- Mathematics, Algebra

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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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

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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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

*Leiba Rodman*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161853
- eISBN:
- 9781400852741
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161853.003.0011
- Subject:
- Mathematics, Algebra

This chapter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution φ. Here, the chapter provides canonical ...
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This chapter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution φ. Here, the chapter provides canonical forms for φ-hermitian pencils, i.e., pencils of the form A + tB, where A and B are both φ-hermitian. It also provides canonical forms for φ-skewhermitian pencils. The canonical forms in question are with respect to either strict equivalence of pencils or to simultaneous φ-congruence of matrices. Applications are made to joint φ-numerical ranges of two φ-skewhermitian matrices and to the corresponding joint φ-numerical cones. The chapter fixes a nonstandard involution φ throughout and a quaternion β(φ) such that φ=(β(φ)) = −β(φ) and ∣β(φ)∣ = 1.Less

This chapter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution φ. Here, the chapter provides canonical forms for φ-hermitian pencils, i.e., pencils of the form *A* + *tB*, where *A* and *B* are both φ-hermitian. It also provides canonical forms for φ-skewhermitian pencils. The canonical forms in question are with respect to either strict equivalence of pencils or to simultaneous φ-congruence of matrices. Applications are made to joint φ-numerical ranges of two φ-skewhermitian matrices and to the corresponding joint φ-numerical cones. The chapter fixes a nonstandard involution φ throughout and a quaternion β(φ) such that φ=(β(φ)) = −β(φ) and ∣β(φ)∣ = 1.

*D. E. Edmunds and W. D. Evans*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0003
- Subject:
- Mathematics, Pure Mathematics

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann ...
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This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.Less

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, *J*-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to *J*-symmetric expressions is proved.