Holdings: Linear algebra and its applications

Linear algebra and its applications:

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Bibliographic Details
Main Author:	Lax, Peter D. 1926- (Author)
Format:	Book
Language:	English
Published:	Hoboken, N.J. Wiley-Interscience 2007
Edition:	2. ed.
Series:	Pure and applied mathematics
Subjects:	Álgebra linear Algebras, Linear Lineare Algebra
Online Access:	Inhaltsverzeichnis Inhaltsverzeichnis
Item Description:	Frühere Aufl. u.d.T.: Linear algebra
Physical Description:	XV, 376 S.
ISBN:	0471751561 9780471751564

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MARC


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adam_text	Contents Preface xi Preface to the First Edition xiii 1. Fundamentals 1 Linear Space, Isomorphism 1 Subspace 4 Linear Dependence 4 Basis, Dimension 5 Quotient Space 8 2. Duality 13 Linear Functions 13 Dual of a Linear Space 14 Annihilator 15 Codimension 16 Quadrature Formula 17 3. Linear Mappings 19 Domain and Target Space 19 Nullspace and Range 20 Fundamental Theorem 20 Underdetermined Linear Systems 21 Interpolation 22 Difference Equations 23 Algebra of Linear Mappings 24 Dimension of Nullspace and Range 27 Transposition 26 Similarity 29 Projections 30 4. Matrices 32 Rows and Columns 33 Matrix Multiplication 35 v Vi CONTENTS Transposition 36 Rank 37 Gaussian Elimination 39 5. Determinant and Trace 44 Ordered Simplices 44 Signed Volume, Determinant 45 Permutation Group 46 Formula for Determinant 48 Multiplicative Property 49 Laplace Expansion 52 Cramer s Rule 54 Trace 55 6. Spectral Theory 58 Iteration of Linear Maps 59 Eigenvalues, Eigenvectors 59 Fibonacci Sequence 62 Characteristic Polynomial 63 Trace and Determinant Revisited 65 Spectral Mapping Theorem 66 Cayley-Hamilton Theorem 67 Generalized Eigenvectors 69 Spectral Theorem 70 Minimal Polynomial 72 When Are Two Matrices Similar 73 Commuting Maps 74 7. Euclidean Structure 77 Scalar Product, Distance 79 Schwarg Inequality 79 Orthonormal Basis 80 Gram-Schmidt 81 Orthogonal Complement 82 Orthogonal Projection 83 Adjoint 84 Overdetermined Systems 86 Isometry 87 The Orthogonal Group 89 Norm of a Linear Map 89 Completeness Local Compactness 92 Complex Euclidean Structure 95 Spectral Radius 97 Hilbert-Schmidt Norm 98 Cross Product 99 CONTENTS Vii 8. Spectral Theory of Self-Adjoint Mappings 101 Quadratic Forms 102 Law of Inertia 103 Spectral Resolution 105 Commuting Maps 111 Anti-Self-Adjoint Maps 112 Normal Maps 112 Rayleigh Quotient 114 Minmax Principle 116 Norm and Eigenvalues 119 9. Calculus of Vector- and Matrix-Valued Functions 121 Convergence in Norm 121 Rules of Differentiation 122 Derivative of det A(t) 126 Matrix Exponential 128 Simple Eigenvalues 129 Multiple Eigenvalues 135 Rellich s Theorem 140 Avoidance of Crossing 140 10. Matrix Inequalities 143 Positive Self-Adjoint Matrices 143 Monotone Matrix Functions 151 Gram Matrices 152 Schur s Theorem 153 The Determinant of Positive Matrices 154 Integral Formula for Determinants 157 Eigenvalues 160 Separation of Eigenvalues 161 Wielandt-Hoffman Theorem 164 Smallest and Largest Eigenvalue 166 Matrices with Positive Self-Adjoint Part 167 Polar Decomposition 169 Singular Values 170 Singular Value Decomposition 170 11. Kinematics and Dynamics 172 Axis and Angle of Rotation 172 Rigid Motion 173 Angular Velocity Vector 176 Fluid Flow 177 Curl and Divergence 179 Small Vibrations 180 Conservation of Energy 182 Frequencies and Normal Modes 184 Viii CONTENTS 12. Convexity 187 Convex Sets 187 Gauge Function 188 Hahn-Banach Theorem 191 Support Function 193 Caratheodory s Theorem 195 Konig-Birkhoff Theorem 198 Helly s Theorem 199 13. The Duality Theorem 202 Farkas-Minkowski Theorem 203 Duality Theorem 206 Economics Interpretation 208 Minmax Theorem 210 14. Normed Linear Spaces 214 Norm 214 F Norms 215 Equivalence of Norms 217 Completeness 219 Local Compactness 219 Theorem of F. Riesz 219 Dual Norm 222 Distance from Subspace 223 Normed Quotient Space 224 Complex Normed Spaces 226 Complex Hahn-Banach Theorem 226 Characterization of Euclidean Spaces 227 15. Linear Mappings Between Normed Linear Spaces 229 Norm of a Mapping 230 Norm of Transpose 231 Normed Algebra of Maps 232 Invertible Maps 233 Spectral Radius 236 16. Positive Matrices 237 Perron s Theorem 237 Stochastic Matrices 240 Frobenius Theorem 243 17. How to Solve Systems of Linear Equations 246 History 246 Condition Number 248 Iterative Methods 248 Steepest Descent 249 Chebychev Iteration 252 CONTENTS ix Three-term Chebychev Iteration 255 Optimal Three-Term Recursion Relation 256 Rate of Convergence 261 18. How to Calculate the Eigenvalues of Self-Adjoint Matrices 262 QR Factorization 262 Using the QR Factorization to Solve Systems of Equations 263 The QR Algorithm for Finding Eigenvalues 263 Householder Reflection for QR Factorization 266 Tridiagonal Form 267 Analogy of QR Algorithm and Toda Flow 269 Moser s Theorem 273 More General Flows 276 19. Solutions 278 Bibliography 300 Appendix 1. Special Determinants 302 Appendix 2. The Pfaffian 305 Appendix 3. Symplectic Matrices 308 Appendix 4. Tensor Product 313 Appendix 5. Lattices 317 Appendix 6. Fast Matrix Multiplication 320 Appendix 7. Gershgorin s Theorem 323 Appendix 8. The Multiplicity of Eigenvalues 325 Appendix 9. The Fast Fourier Transform 328 Appendix 10. The Spectral Radius 334 Appendix 11. The Lorentz Group 342 Appendix 12. Compactness of the Unit Ball 352 Appendix 13. A Characterization of Commutators 355 Appendix 14. Liapunov s Theorem 357 Appendix 15. The Jordan Canonical Form 363 Appendix 16. Numerical Range 367 Index 373
adam_txt	Contents Preface xi Preface to the First Edition xiii 1. Fundamentals 1 Linear Space, Isomorphism 1 Subspace 4 Linear Dependence 4 Basis, Dimension 5 Quotient Space 8 2. Duality 13 Linear Functions 13 Dual of a Linear Space 14 Annihilator 15 Codimension 16 Quadrature Formula 17 3. Linear Mappings 19 Domain and Target Space 19 Nullspace and Range 20 Fundamental Theorem 20 Underdetermined Linear Systems 21 Interpolation 22 Difference Equations 23 Algebra of Linear Mappings 24 Dimension of Nullspace and Range 27 Transposition 26 Similarity 29 Projections 30 4. Matrices 32 Rows and Columns 33 Matrix Multiplication 35 v Vi CONTENTS Transposition 36 Rank 37 Gaussian Elimination 39 5. Determinant and Trace 44 Ordered Simplices 44 Signed Volume, Determinant 45 Permutation Group 46 Formula for Determinant 48 Multiplicative Property 49 Laplace Expansion 52 Cramer's Rule 54 Trace 55 6. Spectral Theory 58 Iteration of Linear Maps 59 Eigenvalues, Eigenvectors 59 Fibonacci Sequence 62 Characteristic Polynomial 63 Trace and Determinant Revisited 65 Spectral Mapping Theorem 66 Cayley-Hamilton Theorem 67 Generalized Eigenvectors 69 Spectral Theorem 70 Minimal Polynomial 72 When Are Two Matrices Similar 73 Commuting Maps 74 7. Euclidean Structure 77 Scalar Product, Distance 79 Schwarg Inequality 79 Orthonormal Basis 80 Gram-Schmidt 81 Orthogonal Complement 82 Orthogonal Projection 83 Adjoint 84 Overdetermined Systems 86 Isometry 87 The Orthogonal Group 89 Norm of a Linear Map 89 Completeness Local Compactness 92 Complex Euclidean Structure 95 Spectral Radius 97 Hilbert-Schmidt Norm 98 Cross Product 99 CONTENTS Vii 8. Spectral Theory of Self-Adjoint Mappings 101 Quadratic Forms 102 Law of Inertia 103 Spectral Resolution 105 Commuting Maps 111 Anti-Self-Adjoint Maps 112 Normal Maps 112 Rayleigh Quotient 114 Minmax Principle 116 Norm and Eigenvalues 119 9. Calculus of Vector- and Matrix-Valued Functions 121 Convergence in Norm 121 Rules of Differentiation 122 Derivative of det A(t) 126 Matrix Exponential 128 Simple Eigenvalues 129 Multiple Eigenvalues 135 Rellich's Theorem 140 Avoidance of Crossing 140 10. Matrix Inequalities 143 Positive Self-Adjoint Matrices 143 Monotone Matrix Functions 151 Gram Matrices 152 Schur's Theorem 153 The Determinant of Positive Matrices 154 Integral Formula for Determinants 157 Eigenvalues 160 Separation of Eigenvalues 161 Wielandt-Hoffman Theorem 164 Smallest and Largest Eigenvalue 166 Matrices with Positive Self-Adjoint Part 167 Polar Decomposition 169 Singular Values 170 Singular Value Decomposition 170 11. Kinematics and Dynamics 172 Axis and Angle of Rotation 172 Rigid Motion 173 Angular Velocity Vector 176 Fluid Flow 177 Curl and Divergence 179 Small Vibrations 180 Conservation of Energy 182 Frequencies and Normal Modes 184 Viii CONTENTS 12. Convexity 187 Convex Sets 187 Gauge Function 188 Hahn-Banach Theorem 191 Support Function 193 Caratheodory's Theorem 195 Konig-Birkhoff Theorem 198 Helly's Theorem 199 13. The Duality Theorem 202 Farkas-Minkowski Theorem 203 Duality Theorem 206 Economics Interpretation 208 Minmax Theorem 210 14. Normed Linear Spaces 214 Norm 214 F Norms 215 Equivalence of Norms 217 Completeness 219 Local Compactness 219 Theorem of F. Riesz 219 Dual Norm 222 Distance from Subspace 223 Normed Quotient Space 224 Complex Normed Spaces 226 Complex Hahn-Banach Theorem 226 Characterization of Euclidean Spaces 227 15. Linear Mappings Between Normed Linear Spaces 229 Norm of a Mapping 230 Norm of Transpose 231 Normed Algebra of Maps 232 Invertible Maps 233 Spectral Radius 236 16. Positive Matrices 237 Perron's Theorem 237 Stochastic Matrices 240 Frobenius' Theorem 243 17. How to Solve Systems of Linear Equations 246 History 246 Condition Number 248 Iterative Methods 248 Steepest Descent 249 Chebychev Iteration 252 CONTENTS ix Three-term Chebychev Iteration 255 Optimal Three-Term Recursion Relation 256 Rate of Convergence 261 18. How to Calculate the Eigenvalues of Self-Adjoint Matrices 262 QR Factorization 262 Using the QR Factorization to Solve Systems of Equations 263 The QR Algorithm for Finding Eigenvalues 263 Householder Reflection for QR Factorization 266 Tridiagonal Form 267 Analogy of QR Algorithm and Toda Flow 269 Moser's Theorem 273 More General Flows 276 19. Solutions 278 Bibliography 300 Appendix 1. Special Determinants 302 Appendix 2. The Pfaffian 305 Appendix 3. Symplectic Matrices 308 Appendix 4. Tensor Product 313 Appendix 5. Lattices 317 Appendix 6. Fast Matrix Multiplication 320 Appendix 7. Gershgorin's Theorem 323 Appendix 8. The Multiplicity of Eigenvalues 325 Appendix 9. The Fast Fourier Transform 328 Appendix 10. The Spectral Radius 334 Appendix 11. The Lorentz Group 342 Appendix 12. Compactness of the Unit Ball 352 Appendix 13. A Characterization of Commutators 355 Appendix 14. Liapunov's Theorem 357 Appendix 15. The Jordan Canonical Form 363 Appendix 16. Numerical Range 367 Index 373
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spelling	Lax, Peter D. 1926- Verfasser (DE-588)130442437 aut Linear algebra and its applications Peter D. Lax 2. ed. Hoboken, N.J. Wiley-Interscience 2007 XV, 376 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics Frühere Aufl. u.d.T.: Linear algebra Álgebra linear larpcal Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s DE-604 http://www.loc.gov/catdir/toc/ecip0719/2007023226.html Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016275059&sequence=000008&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Lax, Peter D. 1926- Linear algebra and its applications Álgebra linear larpcal Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd
subject_GND	(DE-588)4035811-2
title	Linear algebra and its applications
title_auth	Linear algebra and its applications
title_exact_search	Linear algebra and its applications
title_exact_search_txtP	Linear algebra and its applications
title_full	Linear algebra and its applications Peter D. Lax
title_fullStr	Linear algebra and its applications Peter D. Lax
title_full_unstemmed	Linear algebra and its applications Peter D. Lax
title_short	Linear algebra and its applications
title_sort	linear algebra and its applications
topic	Álgebra linear larpcal Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd
topic_facet	Álgebra linear Algebras, Linear Lineare Algebra
url	http://www.loc.gov/catdir/toc/ecip0719/2007023226.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016275059&sequence=000008&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT laxpeterd linearalgebraanditsapplications

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