Matrix mathematics a second course in linear algebra
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| Vorheriger Titel: | Garcia, Stephan Roman A second course in linear algebra |
|---|---|
| Hauptverfasser: | , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore
Cambridge University Press
2023
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Cambridge mathematical textbooks
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 245 | 1 | 0 | |a Matrix mathematics |b a second course in linear algebra |c Stephan Ramon Garcia (Pomona College), Roger A. Horn (Tampa, Florida) |
| 246 | 1 | 3 | |a Second course in linear algebra |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore |b Cambridge University Press |c 2023 | |
| 264 | 4 | |c © 2023 | |
| 300 | |a xxi, 467 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge mathematical textbooks | |
| 500 | |a Literaturverzeichnis Seite 449. - Index Seite 450-467 | ||
| 500 | |a Erste Auflage erschien unter dem Titel: A second course in linear algebra, 2017 | ||
| 500 | |a 2305 | ||
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Horn, Roger A. |d 1942- |e Verfasser |0 (DE-588)142437808 |4 aut | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202 MAT 150 2023 B 1737(2) |
|---|---|
| DE-BY-TUM_katkey | 2758636 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040009553662 |
| _version_ | 1820870223806332928 |
| adam_text | Preface for the Second Edition List ofNotation 1 Vector Spaces 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2 What Is a Basis? Dimension Full-Rank Factorizations Coordinate Vectors and Matrix Representations of Linear Transformations Change of Basis Similarity Polynomial Bases and Lagrange Interpolation Problems Notes Some Important Concepts Block Matrices 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4 What Is a Vector Space? Examples of Vector Spaces Subspaces Linear Combinations, Lists, and Span Intersections, Sums, and Direct Sums of Subspaces Linear Dependence and Linear Independence The Pivot Column Decomposition Problems Notes Some Important Concepts Bases and Similarity 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 3 page xiii xviii Row and Column Partitions Block Partitions and Direct Sums Determinants of Block Matrices Kronecker Products Problems Notes Some Important Concepts Rank, Triangular Factorizations, and Row Equivalence 4.1 The Rank-Nullity Theorem and Subspace Intersection 1 1 3 5 7 11 13 18 20 24 24 26 26 28 31 33 38 40 43 49 55 55 56 56 61 65 68 70 73 73 74 74
viii Contents 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Rank LU Factorization Row Equivalence Commutators and Shoda’s Theorem Problems Notes Some Important Concepts Products and Norms The Pythagorean Theorem and the Law of Cosines Angles and Lengths in the Plane Inner Products The Norm Derived from an Inner Product Normed Vector Spaces Problems Notes Some Important Concepts 78 80 83 85 87 92 93 5 Inner 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6 Orthonormal Vectors 6.1 Orthonormal Sequences 6.2 Orthonormal Bases 6.3 The Gram—Schmidt Process 6.4 The Riesz Representation Theorem 6.5 Orthonormal Bases and Linear Transformations 6.6 Adjoints of Linear Transformations and Matrices 6.7 Parseval’s Identity and Bessel’s Inequality 6.8 Fourier Series 6.9 Orthogonal Polynomial Bases and Gaussian Quadrature 6.10 Problems 6.11 Notes 6.12 Some Important Concepts ill 111 113 114 117 118 119 122 124 128 132 136 136 7 Unitary Matrices 7.1 Unitary Matrices 7.2 Change of Orthonormal Basis and Unitary Similarity 7.3 Permutation Matrices 7.4 The 127? Factorization 7.5 Upper Hessenberg Matrices 7.6 Problems ΊΠ Notes 7.8 Some Important Concepts 137 137 143 145 147 151 153 157 157 8 Orthogonal Complements and Orthogonal Projections 8.1 Orthogonal Complements 8.2 The Minimum-Norm Solution of a Consistent Linear System 8.3 Orthogonal Projections 158 158 160 163 94 94 95 98 101 106 107 110 110
Contents 8.4 8.5 8.6 8.7 8.8 8.9 9 Best Approximation The Least-Squares Solution of an Inconsistent Linear System Invariant Subspaces Problems Notes Some Important Concepts Eigenvalues, Eigenvectors, and Geometric Multiplicity 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Eigenvalue-Eigenvector Pairs Every Square Complex Matrix Has an Eigenvalue How Many Eigenvalues Are There? The Eigenvalues Are in Gershgorin Disks Eigenvectors and Commuting Matrices Real Similarity of Real Matrices Problems Notes Some Important Concepts 10 The Characteristic Polynomial and Algebraic Multiplicity 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 11 The Characteristic Polynomial Algebraic Multiplicity Similarity and Eigenvalue Multiplicities Diagonalization and Eigenvalue Multiplicities The Functional Calculus for Diagonalizable Matrices Commutants The Eigenvalues of AB and BA Problems Notes Some Important Concepts Unitary Triangularization and Block Diagonalization 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 Schur’s Triangularization Theorem The Cayley—Hamilton Theorem The Minimal Polynomial Linear Matrix Equations and Block Diagonalization Commuting Matrices and Triangularization Eigenvalue Adjustments and the Google Matrix Problems Notes Some Important Concepts 12 The Jordan Form: Existence and Uniqueness 12.1 12.2 12.3 12.4 Ranks of Powers Jordan Blocks and Jordan Matrices Existence of a Jordan Form Uniqueness of a Jordan Form ix 167 171 174 176 179 179 180 180 184 186 190 195 197 198 201 201 202 202 204 206 207 211 213 214 216 220 220 221 221 223 225 228 231 233 234 237 238 239 239 240 243 246
Contents x The Jordan Canonical Form Problems Notes Some Important Concepts 250 251 254 255 The Jordan Form: Applications 256 256 258 261 262 267 269 270 271 276 276 12.5 12.6 12.7 12.8 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 Differential Equations and the Jordan Canonical Form Convergent and Power-Bounded Matrices The Jordan Forms of A and Ap Stochastic Matrices The Invertible Jordan Blocks of AB and BA Similarity of a Matrix and Its Transpose Similarity of a Matrix and Its Complex Conjugate Problems Notes Some Important Concepts Normal Matrices and the Spectral Theorem 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 Normal Matrices The Spectral Theorem The Defect from Normality The Fuglede-Putnam Theorem Circulant, Fourier, and Hartley Matrices Some Special Classes of Normal Matrices Similarity of Normal and Other Diagonalizable Matrices Some Characterizations of Normality Spectral Resolutions Problems Notes Some Important Concepts Positive Semidefinite Matrices 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 Positive Semidefinite Matrices Schur Complements and Diagonal Dominance The Square Root of a Positive Semidefinite Matrix The Cholesky Factorization Simultaneous Diagonalization of Quadratic Forms The Schur Product Theorem Problems Notes Some Important Concepts The Singular Value and Polar Decompositions 16.1 16.2 16.3 The Singular Value Decomposition The Compact Singular Value Decomposition The Polar Decomposition 277 277 279 282 283 284 287 290 291 292 296 300 300 301 301 306 308 311 313 315 317 324 325 326 326 331 333
Contents 16.4 16.5 16.6 16.7 Unitary Equivalence and Bidiagonal Matrices Problems Notes Some Important Concepts 17 Singular Values and the Spectral Norm 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 Singular Values and Approximations The Spectral and Frobenius Norms Singular Values and Eigenvalues The Pseudoinverse The Spectral Condition Number Complex Symmetric Matrices Idempotent Matrices Problems Notes Some Important Concepts xi 337 339 342 343 344 344 346 349 353 358 361 362 364 368 368 18 Interlacing and Inertia 18.1 The Rayleigh Quotient 18.2 Eigenvalue Interlacing for Sums of Hermitian Matrices 18.3 Eigenvalue Interlacing for Bordered Hermitian Matrices 18.4 Sylvester’s Criterion 18.5 Diagonal Entries and Eigenvalues of Hermitian Matrices 18.6 *Congruence and Inertia of Hermitian Matrices 18.7 Weyl’s Inequalities 18.8 *Congruence and Inertia of Normal Matrices 18.9 Problems 18.10 Notes 18.11 Some Important Concepts 369 369 370 372 375 376 377 379 381 383 388 389 19 Norms and Matrix Norms 19.1 Norms of Vectors 19.2 Norms of Matrices 19.3 Induced Matrix Norms 19.4 Matrix Norms and the Spectral Radius 19.5 The Point Jacobi Iterative Method 19.6 The Power Method for a Dominant Eigenpair 19.7 Problems 19.8 Notes 19.9 Some Important Concepts 390 390 391 393 395 399 400 402 404 405 20 Positive and Nonnegative Matrices 20.1 Nonnegative Matrices 406 406 410 411 412 20.2 20.3 20.4 Positive Matrices Primitive Matrices The Power Method for Primitive Matrices
Contents xii 20.5 20.6 20.7 Problems Notes Some Important Concepts 413 415 416 Appendix A Complex Numbers A.l Complex Numbers A.2 Modulus and Argument A.3 Conjugation and Complex Arithmetic A.4 Polar Form of a Complex Number A.5 The Complex Exponential Function A. 6 Problems Appendix В Polynomials B.l Polynomials В.2 The Division Algorithm B.3 Zeros and Roots B.4 Identity Theorems for Polynomials B.5 Polynomials and Matrices B.6 Problems Appendix C Basic Linear Algebra C.l Functions and Sets C.2 Matrices C.3 Systems of Linear Equations C.4 Determinants C.5 Problems Appendix D Induction D.l Mathematical Induction D.2 Problems 417 417 420 420 423 425 426 428 428 428 429 429 429 430 432 432 433 439 441 444 447 447 448 References Index 449 450
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| any_adam_object | 1 |
| author | Garcia, Stephan Ramon Horn, Roger A. 1942- |
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| author_facet | Garcia, Stephan Ramon Horn, Roger A. 1942- |
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| author_sort | Garcia, Stephan Ramon |
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| building | Verbundindex |
| bvnumber | BV048994127 |
| classification_rvk | SK 220 |
| classification_tum | MAT 150 |
| ctrlnum | (OCoLC)1389177345 (DE-599)KXP1832534235 |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV048994127 |
| illustrated | Illustrated |
| indexdate | 2024-12-24T09:46:46Z |
| institution | BVB |
| isbn | 9781108837101 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034257419 |
| oclc_num | 1389177345 |
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| owner_facet | DE-739 DE-91G DE-BY-TUM DE-83 DE-523 DE-11 DE-20 DE-473 DE-BY-UBG |
| physical | xxi, 467 Seiten Illustrationen, Diagramme |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | Cambridge mathematical textbooks |
| spellingShingle | Garcia, Stephan Ramon Horn, Roger A. 1942- Matrix mathematics a second course in linear algebra Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Matrix mathematics a second course in linear algebra |
| title_alt | Second course in linear algebra |
| title_auth | Matrix mathematics a second course in linear algebra |
| title_exact_search | Matrix mathematics a second course in linear algebra |
| title_full | Matrix mathematics a second course in linear algebra Stephan Ramon Garcia (Pomona College), Roger A. Horn (Tampa, Florida) |
| title_fullStr | Matrix mathematics a second course in linear algebra Stephan Ramon Garcia (Pomona College), Roger A. Horn (Tampa, Florida) |
| title_full_unstemmed | Matrix mathematics a second course in linear algebra Stephan Ramon Garcia (Pomona College), Roger A. Horn (Tampa, Florida) |
| title_old | Garcia, Stephan Roman A second course in linear algebra |
| title_short | Matrix mathematics |
| title_sort | matrix mathematics a second course in linear algebra |
| title_sub | a second course in linear algebra |
| topic | Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034257419&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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