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E-raamat: Topics in Quaternion Linear Algebra

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 It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and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

Arvustused

One of Choice's Outstanding Academic Titles for 2015 "Rodman fills a void in the monographic literature with this work."--Choice "The book is self-contained and well organized... Full and detailed proofs are supplied. Another exciting point is the presence of many open problems throughout the book."--Gisele C. Ducati, MatchSciNet

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Short-listed for Choice Magazine Outstanding Reference/Academic Book Award 2015.
Preface xi
1 Introduction
1(8)
1.1 Notation and conventions
5(2)
1.2 Standard matrices
7(2)
2 The algebra of quaternions
9(19)
2.1 Basic definitions and properties
9(2)
2.2 Real linear transformations and equations
11(3)
2.3 The Sylvester equation
14(3)
2.4 Automorphisms and involutions
17(4)
2.5 Quadratic maps
21(2)
2.6 Real and complex matrix representations
23(1)
2.7 Exercises
24(2)
2.8 Notes
26(2)
3 Vector spaces and matrices: Basic theory
28(36)
3.1 Finite dimensional quaternion vector spaces
28(2)
3.2 Matrix algebra
30(3)
3.3 Real matrix representation of quaternions
33(3)
3.4 Complex matrix representation of quaternions
36(2)
3.5 Numerical ranges with respect to conjugation
38(6)
3.6 Matrix decompositions: nonstandard involutions
44(3)
3.7 Numerical ranges with respect to nonstandard involutions
47(5)
3.8 Proof of Theorem 3.7.5
52(4)
3.9 The metric space of subspaces
56(3)
3.10 Appendix: Multivariable real analysis
59(2)
3.11 Exercises
61(2)
3.12 Notes
63(1)
4 Symmetric matrices and congruence
64(19)
4.1 Canonical forms under congruence
64(5)
4.2 Neutral and semidefinite subspaces
69(3)
4.3 Proof of Theorem 4.2.6
72(3)
4.4 Proof of Theorem 4.2.7
75(3)
4.5 Representation of semidefinite subspaces
78(2)
4.6 Exercises
80(2)
4.7 Notes
82(1)
5 Invariant subspaces and Jordan form
83(48)
5.1 Root subspaces
83(2)
5.2 Root subspaces and matrix representations
85(5)
5.3 Eigenvalues and eigenvectors
90(4)
5.4 Some properties of Jordan blocks
94(3)
5.5 Jordan form
97(5)
5.6 Proof of Theorem 5.5.3
102(7)
5.7 Jordan forms of matrix representations
109(2)
5.8 Comparison with real and complex similarity
111(2)
5.9 Determinants
113(2)
5.10 Determinants based on real matrix representations
115(1)
5.11 Linear matrix equations
116(3)
5.12 Companion matrices and polynomial equations
119(4)
5.13 Eigenvalues of hermitian matrices
123(1)
5.14 Differential and difference equations
123(3)
5.15 Appendix: Continuous roots of polynomials
126(1)
5.16 Exercises
127(3)
5.17 Notes
130(1)
6 Invariant neutral and semidefinite subspaces
131(22)
6.1 Structured matrices and invariant neutral subspaces
132(4)
6.2 Invariant semidefinite subspaces respecting conjugation
136(3)
6.3 Proof of Theorem 6.2.6
139(4)
6.4 Unitary, dissipative, and expansive matrices
143(3)
6.5 Invariant semidefinite subspaces: Nonstandard involution
146(2)
6.6 Appendix: Convex sets
148(1)
6.7 Exercises
149(2)
6.8 Notes
151(2)
7 Smith form and Kronecker canonical form
153(19)
7.1 Matrix polynomials with quaternion coefficients
153(5)
7.2 Nonuniqueness of the Smith form
158(3)
7.3 Statement of the Kronecker form
161(2)
7.4 Proof of Theorem 7.3.2: Existence
163(4)
7.5 Proof of Theorem 7.3.2: Uniqueness
167(2)
7.6 Comparison with real and complex strict equivalence
169(1)
7.7 Exercises
170(1)
7.8 Notes
171(1)
8 Pencils of hermitian matrices
172(22)
8.1 Canonical forms
172(5)
8.2 Proof of Theorem 8.1.2
177(4)
8.3 Positive semidefinite linear combinations
181(2)
8.4 Proof of Theorem 8.3.3
183(4)
8.5 Comparison with real and complex congruence
187(1)
8.6 Expansive and plus-matrices: Singular H
188(3)
8.7 Exercises
191(1)
8.8 Notes
192(2)
9 Skewhermitian and mixed pencils
194(34)
9.1 Canonical forms for skewhermitian matrix pencils
194(3)
9.2 Comparison with real and complex skewhermitian pencils
197(2)
9.3 Canonical forms for mixed pencils: Strict equivalence
199(3)
9.4 Canonical forms for mixed pencils: Congruence
202(3)
9.5 Proof of Theorem 9.4.1: Existence
205(5)
9.6 Proof of Theorem 9.4.1: Uniqueness
210(5)
9.7 Comparison with real and complex pencils: Strict equivalence
215(4)
9.8 Comparison with complex pencils: Congruence
219(2)
9.9 Proofs of Theorems 9.7.2 and 9.8.1
221(3)
9.10 Canonical forms for matrices under congruence
224(2)
9.11 Exercises
226(1)
9.12 Notes
227(1)
10 Indefinite inner products: Conjugation
228(33)
10.1 H-hermitian and H-skewhermitian matrices
229(3)
10.2 Invariant semidefinite subspaces
232(3)
10.3 Invariant Lagrangian subspaces I
235(3)
10.4 Differential equations I
238(4)
10.5 Hamiltonian, skew-Hamiltonian matrices: Canonical forms
242(4)
10.6 Invariant Lagrangian subspaces II
246(2)
10.7 Extension of subspaces
248(2)
10.8 Proofs of Theorems 10.7.2 and 10.7.5
250(5)
10.9 Differential equations II
255(2)
10.10 Exercises
257(2)
10.11 Notes
259(2)
11 Matrix pencils with symmetries: Nonstandard involution
261(18)
11.1 Canonical forms for ø-hermitian pencils
261(2)
11.2 Canonical forms for ø-skewhermitian pencils
263(3)
11.3 Proof of Theorem 11.2.2
266(8)
11.4 Numerical ranges and cones
274(3)
11.5 Exercises
277(1)
11.6 Notes
278(1)
12 Mixed matrix pencils: Nonstandard involutions
279(21)
12.1 Canonical forms for ø-mixed pencils: Strict equivalence
279(2)
12.2 Proof of Theorem 12.1.2
281(3)
12.3 Canonical forms of ø-mixed pencils: Congruence
284(3)
12.4 Proof of Theorem 12.3.1
287(3)
12.5 Strict equivalence versus ø-congruence
290(1)
12.6 Canonical forms of matrices under ø-congruence
291(1)
12.7 Comparison with real and complex matrices
292(2)
12.8 Proof of Theorem 12.7.4
294(4)
12.9 Exercises
298(1)
12.10 Notes
299(1)
13 Indefinite inner products: Nonstandard involution
300(28)
13.1 Canonical forms: Symmetric inner products
301(5)
13.2 Canonical forms: Skewsymmetric inner products
306(3)
13.3 Extension of invariant semidefinite subspaces
309(4)
13.4 Proofs of Theorems 13.3.3 and 13.3.4
313(3)
13.5 Invariant Lagrangian subspaces
316(5)
13.6 Boundedness of solutions of differential equations
321(4)
13.7 Exercises
325(2)
13.8 Notes
327(1)
14 Matrix equations
328(11)
14.1 Polynomial equations
328(3)
14.2 Bilateral quadratic equations
331(1)
14.3 Algebraic Riccati equations
332(5)
14.4 Exercises
337(1)
14.5 Notes
338(1)
15 Appendix: Real and complex canonical forms
339(14)
15.1 Jordan and Kronecker canonical forms
339(2)
15.2 Real matrix pencils with symmetries
341(7)
15.3 Complex matrix pencils with symmetries
348(5)
Bibliography 353(8)
Index 361
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.
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