Linear algebra in action:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, R.I.
American Mathematical Society
2013
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate studies in mathematics
78 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XIX, 585 S. graph. Darst. |
| ISBN: | 9781470409081 |
Internformat
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| 650 | 7 | |a Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 7 | |a Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 7 | |a Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 7 | |a Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 7 | |a Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 4 | |a Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) | |
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Datensatz im Suchindex
| DE-BY-862_location | 2000 |
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| adam_text | Titel: Linear algebra in action
Autor: Dym, Harry
Jahr: 2013
Contents
Preface to the Second Edition xv
Preface to the First Edition xvii
Chapter 1. Vector spaces 1
§1.1. Preview 1
§1.2. The abstract definition of a vector space 2
§1.3. Some definitions 5
§1.4. Mappings 11
§1.5. Triangular matrices 13
§1.6. Block triangular matrices 17
§1.7. Schur complements 18
§1.8. Other matrix products 20
Chapter 2. Gaussian elimination 21
§2.1. Some preliminary observations 22
§2.2. Examples 24
§2.3. Upper echelon matrices 30
§2.4. The conservation of dimension 36
§2.5. Quotient spaces 38
§2.6. Conservation of dimension for matrices 38
§2.7. From U to A 40
§2.8. Square matrices 41
Chapter 3. Additional applications of Gaussian elimination 45
v
vi
Contents
§3.1. Gaussian elimination redux 45
§3.2. Properties of BA and AC 48
§3.3. Extracting a basis 50
§3.4. Computing the coefficients in a basis 51
§3.5. The Gauss-Seidel method 52
§3.6. Block Gaussian elimination 55
§3.7. {0, 1, oo} 56
§3.8. Review 57
Chapter 4. Eigenvalues and eigenvectors 61
§4.1. Change of basis and similarity 62
§4.2. Invariant subspaces 64
§4.3. Existence of eigenvalues 64
§4.4. Eigenvalues for matrices 66
§4.5. Direct sums 69
§4.6. Diagonalizable matrices 72
§4.7. An algorithm for diagonalizing matrices 74
§4.8. Computing eigenvalues at this point 75
§4.9. Not all matrices are diagonalizable 77
§4.10. The Jordan decomposition theorem 80
§4.11. An instructive example 81
§4.12. The binomial formula 83
§4.13. More direct sum decompositions 84
§4.14. Verification of Theorem 4.13 87
§4.15. Bibliographical notes 89
Chapter 5. Determinants 91
§5.1. Functional 91
§5.2. Determinants 92
§5.3. Useful rules for calculating determinants 95
§5.4. Eigenvalues 98
§5.5. Exploiting block structure 101
§5.6. The Binet-Cauchy formula 103
§5.7. Minors 105
§5.8. Uses of determinants 109
§5.9. Companion matrices 110
Contents
vii
§5.10. Circulants and Vandermonde matrices 111
Chapter 6. Calculating Jordan forms 113
§6.1. Overview 114
§6.2. Structure of the nullspaees Afgj 114
§6.3. Chains and cells 116
§6.4. Computing J 117
§6.5. An algorithm for computing U 118
§6.6. A simple example 120
§6.7. A more elaborate example 122
§6.8. Jordan decompositions for real matrices 125
§6.9. Projection matrices 128
§6.10. Companion and generalized Vandermonde matrices 128
Chapter 7. Normed linear spaces 133
§7.1. Four inequalities 133
§7.2. Normed linear spaces 138
§7.3. Equivalence of norms 140
§7.4. Norms of linear transformations 142
§7.5. Operator norms for matrices 144
§7.6. Mixing tops and bottoms 146
§7.7. Evaluating some operator norms 146
§7.8. Inequalities for multiplicative norms 148
§7.9. Small perturbations 151
§7.10. Bounded linear functionals 154
§7.11. Extensions of bounded linear functionals 155
§7.12. Banach spaces 158
§7.13. Bibliographical notes 160
Chapter 8. Inner product spaces and orthogonality 161
§8.1. Inner product spaces 161
§8.2. A characterization of inner product spaces 164
§8.3. Orthogonality 165
§8.4. Gram matrices 167
§8.5. Projections and direct sum decompositions 168
§8.6. Orthogonal projections 170
§8.7. Orthogonal expansions 173
viii
Contents
§8.8. The Gram-Schmidt method 175
§8.9. Toeplitz and Hankel matrices 176
§8.10. Adjoints 178
§8.11. The Riesz representation theorem 182
§8.12. Normal, selfadjoint and unitary transformations 184
§8.13. Auxiliary formulas 186
§8.14. Gaussian quadrature 187
§8.15. Bibliographical notes 190
Chapter 9. Symmetric, Hermitian and normal matrices 191
§9.1. Hermitian matrices are diagonalizable 192
§9.2. Commuting Hermitian matrices 194
§9.3. Real Hermitian matrices 196
§9.4. Projections and direct sums in F™ 197
§9.5. Projections and rank 201
§9.6. Normal matrices 202
§9.7. QR factorization 204
§9.8. Schur s theorem 205
§9.9. Areas, volumes and determinants 207
§9.10. Boundary value problems 212
§9.11. Bibliographical notes 212
Chapter 10. Singular values and related inequalities 213
§10.1. Singular value decompositions 213
§10.2. Complex symmetric matrices 218
§10.3. Approximate solutions of linear equations 220
§10.4. Fitting a line in M2 221
§10.5. Fitting a line in Rp 222
§10.6. Projection by iteration 223
§10.7. The Courant-Fischer theorem 224
§10.8. Inequalities for singular values 228
§10.9. von Neumann s inequality for contractive matrices 235
§10.10. Bibliographical notes 236
Chapter 11. Pseudoinverses 237
§11.1. Pseudoinverses 237
§11.2. The Moore-Penrose inverse 244
Contents
ix
§11.3. Best approximation in terms of Moore-Penrose inverses 247
§11.4. Drazin inverses 249
§11.5. Bibliographical notes 250
Chapter 12. Triangular factorization and positive definite matrices 251
§12.1. A detour on triangular factorization 252
§12.2. Definite and semidefinite matrices 254
§12.3. Characterizations of positive definite matrices 256
§12.4. An application of factorization 259
§12.5. Positive definite Toeplitz matrices 260
§12.6. Detour on block Toeplitz matrices 266
§12.7. A maximum entropy matrix completion problem 271
§12.8. A class of A - O for which (12.52) holds 275
§12.9. Schur complements for semidefinite matrices 277
§12.10. Square roots 280
§12.11. Polar forms 282
§12.12. Matrix inequalities 283
§12.13. A minimal norm completion problem 286
§12.14. A description of all solutions to the minimal norm
completion problem 288
§12.15. Bibliographical notes 289
Chapter 13. Difference equations and differential equations 291
§13.1. Systems of difference equations 292
§13.2. Nonhomogeneous systems of difference equations 293
§13.3. The exponential etA 294
§13.4. Systems of differential equations 296
§13.5. Uniqueness 298
§13.6. Isometric and isospectral flows 299
§13.7. Second-order differential systems 300
§13.8. Stability 301
§13.9. Nonhomogeneous differential systems 301
§13.10. Strategy for equations 302
§13.11. Second-order difference equations 303
§13.12. Higher order difference equations 306
§13.13. Second-order differential equations 307
§13.14. Higher order differential equations 309
X
Contents
§13.15. Wronskians 311
§13.16. Variation of parameters 313
Chapter 14. Vector-valued functions 315
§14.1. Mean value theorems 315
§14.2. Taylor s formula with remainder 316
§14.3. Application of Taylor s formula with remainder 317
§14.4. Mean value theorem for functions of several variables 318
§14.5. Mean value theorems for vector-valued functions of several
variables 319
§14.6. A contractive fixed point theorem 321
§14.7. Newton s method 324
§14.8. A refined contractive fixed point theorem 327
§14.9. Spectral radius 328
§14.10. The Brouwer fixed point theorem 332
§14.11. Bibliographical notes 336
Chapter 15. The implicit function theorem 337
§15.1. Preliminary discussion 337
§15.2. The implicit function theorem 339
§15.3. A generalization of the implicit function theorem 344
§15.4. Continuous dependence of solutions 346
§15.5. The inverse function theorem 347
§15.6. Roots of polynomials 349
§15.7. An instructive example 349
§15.8. A more sophisticated approach 351
§15.9. Dynamical systems 353
§15.10. Lyapunov functions 355
§15.11. Bibliographical notes 357
Chapter 16. Extremal problems 359
§16.1. Classical extremal problems 359
§16.2. Convex functions 363
§16.3. Extremal problems with constraints 366
§16.4. Examples 368
§16.5. Krylov subspaces 374
§16.6. The conjugate gradient method 374
Contents
xi
§16.7. Dual extremal problems 379
§16.8. Linear programming 381
§16.9. Bibliographical notes 386
Chapter 17. Matrix-valued holomorphic functions 387
§17.1. Differentiation 387
§17.2. Contour integration 391
§17.3. Evaluating integrals by contour integration 396
§17.4. A short detour on Fourier analysis 400
§17.5. The Hilbert matrix 403
§17.6. Contour integrals of matrix-valued functions 404
§17.7. Continuous dependence of the eigenvalues 407
§17.8. More on small perturbations 408
§17.9. Spectral radius redux 410
§17.10. Fractional powers 413
§17.11. Bibliographical notes 414
Chapter 18. Matrix equations 415
§18.1. The equation X — AXB = O 415
§18.2. The Sylvester equation AX — XB = C 418
§18.3. AX = XB 421
§18.4. Special classes of solutions 422
§18.5. Riccati equations 424
§18.6. Two lemmas 430
§18.7. An LQR problem 432
§18.8. Bibliographical notes 434
Chapter 19. Realization theory 435
§19.1. Minimal realizations 442
§19.2. Stabilizable and detectable realizations 449
§19.3. Reproducing kernel Hilbert spaces 450
§19.4. de Branges spaces 453
§19.5. Ra invariance 455
§19.6. A left tangential Nevanlinna-Pick interpolation problem 456
§19.7. Factorization of 0(A) 462
§19.8. Bibliographical notes 465
Chapter 20. Eigenvalue location problems 467
Xll
Contents
§20.1. Interlacing 467
§20.2. Sylvester s law of inertia 470
§20.3. Congruence 472
§20.4. Counting positive and negative eigenvalues 474
§20.5. Exploiting continuity 477
§20.6. Gersgorin disks 478
§20.7. The spectral mapping principle 480
§20.8. Inertia theorems 480
§20.9. An eigenvalue assignment problem 483
§20.10. Bibliographical notes 486
Chapter 21. Zero location problems 487
§21.1. Bezoutians 487
§21.2. The Barnett identity 492
§21.3. The main theorem on Bezoutians 493
§21.4. Resultants 495
§21.5. Other directions 499
§21.6. Bezoutians for real polynomials 501
§21.7. Stable polynomials 502
§21.8. Kharitonov s theorem 504
§21.9. Bibliographical notes 505
Chapter 22. Convexity 507
§22.1. Preliminaries 507
§22.2. Convex functions 509
§22.3. Convex sets in R 512
§22.4. Separation theorems in Mn 513
§22.5. Hyperplanes 515
§22.6. Support hyperplanes 516
§22.7. Convex hulls 517
§22.8. Extreme points 520
§22.9. Brouwer s theorem for compact convex sets 522
§22.10. The Minkowski functional 523
§22.11. The numerical range 525
§22.12. Eigenvalues versus numerical range 528
§22.13. The Gauss-Lucas theorem 529
Contents
xiii
§22.14. The Heinz inequality 530
§22.15. Extreme points for polyhedra 532
§22.16. Bibliographical notes 536
Chapter 23. Matrices with nonnegative entries 537
§23.1. Perron-Frobenius theory 538
§23.2. Stochastic matrices 544
§23.3. Behind Google 545
§23.4. Doubly stochastic matrices 546
§23.5. An inequality of Ky Fan 550
§23.6. The Schur-Horn convexity theorem 552
§23.7. Bibliographical notes 558
Appendix A. Some facts from analysis 559
§A.l. Convergence of sequences of points 559
§A.2. Convergence of sequences of functions 560
§A.3. Convergence of sums 560
§A.4. Sups and infs 561
§A.5. Topology 562
§A.6. Compact sets 562
§A.7. Normed linear spaces 562
Appendix B. More complex variables 565
§B.l. Power series 565
§B.2. Isolated zeros 567
§B.3. The maximum modulus principle 569
§B.4. ln(l — A) when |A| 1 569
§B.5. Rouché s theorem 570
§B.6. Liouville s theorem 572
§B.7. Laurent expansions 572
§B.8. Partial fraction expansions 573
Bibliography 575
Notation Index 579
Subject Index 581
|
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.5 |
| dewey-search | 512/.5 |
| dewey-sort | 3512 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
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| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spellingShingle | Dym, Harry 1938- Linear algebra in action Graduate studies in mathematics Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) msc Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) msc Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) msc Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) msc Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) msc Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) msc Algebras, Linear Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Linear algebra in action |
| title_auth | Linear algebra in action |
| title_exact_search | Linear algebra in action |
| title_full | Linear algebra in action Harry Dym |
| title_fullStr | Linear algebra in action Harry Dym |
| title_full_unstemmed | Linear algebra in action Harry Dym |
| title_short | Linear algebra in action |
| title_sort | linear algebra in action |
| topic | Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) msc Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) msc Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) msc Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) msc Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) msc Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) msc Algebras, Linear Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Linear and multilinear algebra; matrix theory ... Instructional exposition (textbooks, tutorial papers, etc.) Functions of a complex variable ... Instructional exposition (textbooks, tutorial papers, etc.) Ordinary differential equations ... Instructional exposition (textbooks, tutorial papers, etc.) Difference and functional equations ... Instructional exposition (textbooks, tutorial papers, etc.) Convex and discrete geometry ... Instructional exposition (textbooks, tutorial papers, etc.) Systems theory; control ... Instructional exposition (textbooks, tutorial papers, etc.) Algebras, Linear Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027016114&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT dymharry linearalgebrainaction |
Inhaltsverzeichnis
THWS Schweinfurt Zentralbibliothek Lesesaal
| Signatur: |
2000 SK 220 D997(2) |
|---|---|
| Exemplar 1 | ausleihbar Checked out – Rückgabe bis: 30.10.2025 Vormerken |