Matrices: theory and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
New York [u.a.]
Springer
2010
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
216 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 289 S. |
| ISBN: | 9781441976826 9781441976833 |
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Datensatz im Suchindex
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| adam_text | IMAGE 1
CONTENTS
PREFACE FOR THE SECOND EDITION . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . XI
PREFACE FOR THE FIRST EDITION . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . XIII
1 ELEMENTARY LINEAR AND MULTILINEAR ALGEBRA . . . . . . . . . . . . . .
. . . . . . . . 1
1.1 VECTORS AND SCALARS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1
1.2 LINEAR MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 5
1.3 BILINEAR MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 9
2 WHAT ARE MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 15
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 15
2.2 MATRICES AS LINEAR MAPS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 19
2.3 MATRICES AND BILINEAR FORMS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 28
3 SQUARE MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 31
3.1 DETERMINANT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 31
3.2 MINORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 34
3.3 INVERTIBILITY . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 38
3.4 EIGENVALUES AND EIGENVECTORS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 42
3.5 THE CHARACTERISTIC POLYNOMIAL . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 43
3.6 DIAGONALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 48
3.7 TRIGONALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 49
3.8 RANK-ONE PERTURBATIONS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 52
3.9 ALTERNATE MATRICES AND THE PFAFFIAN . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 54
3.10 CALCULATING THE CHARACTERISTIC POLYNOMIAL . . . . . . . . . . . . .
. . . . . . . . . 56
3.11 IRREDUCIBLE MATRICES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 59
4 TENSOR AND EXTERIOR PRODUCTS . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 69
4.1 TENSOR PRODUCT OF VECTOR SPACES . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 69
4.2 EXTERIOR CALCULUS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 72
4.3 TENSORIZATION OF LINEAR MAPS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 77
4.4 A POLYNOMIAL IDENTITY IN M N ( K ) . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 78
VII
IMAGE 2
VIII CONTENTS
5 MATRICES WITH REAL OR COMPLEX ENTRIES . . . . . . . . . . . . . . . .
. . . . . . . . . . . 83
5.1 SPECIAL MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 83
5.2 EIGENVALUES OF REAL- AND COMPLEX-VALUED MATRICES . . . . . . . . . .
. . . . 86 5.3 SPECTRAL DECOMPOSITION OF NORMAL MATRICES . . . . . . . .
. . . . . . . . . . . . 91
5.4 NORMAL AND SYMMETRIC REAL-VALUED MATRICES . . . . . . . . . . . . .
. . . . . . 92
5.5 FUNCTIONAL CALCULUS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 94
5.6 NUMERICAL RANGE . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 98
5.7 THE GERSHGORIN DOMAIN . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 102
6 HERMITIAN MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 109
6.1 THE SQUARE ROOT OVER HPD N . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 109
6.2 RAYLEIGH QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 110
6.3 FURTHER PROPERTIES OF THE SQUARE ROOT . . . . . . . . . . . . . . .
. . . . . . . . . . . 114
6.4 SPECTRUM OF RESTRICTIONS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 115
6.5 SPECTRUM VERSUS DIAGONAL . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 117
6.6 THE DETERMINANT OF NONNEGATIVE HERMITIAN MATRICES . . . . . . . . .
. . . . 119
7 NORMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 127
7.1 A BRIEF REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 127
7.2 HOUSEHOLDER S THEOREM . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 133
7.3 AN INTERPOLATION INEQUALITY . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 135
7.4 VON NEUMANN S INEQUALITY . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 137
8 NONNEGATIVE MATRICES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 149
8.1 NONNEGATIVE VECTORS AND MATRICES . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 149
8.2 THE PERRON-FROBENIUS THEOREM: WEAK FORM . . . . . . . . . . . . . .
. . . . . 150
8.3 THE PERRON-FROBENIUS THEOREM: STRONG FORM . . . . . . . . . . . . .
. . . . . 151
8.4 CYCLIC MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 154
8.5 STOCHASTIC MATRICES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 156
9 MATRICES WITH ENTRIES IN A PRINCIPAL IDEAL DOMAIN; JORDAN REDUCTION
163 9.1 RINGS, PRINCIPAL IDEAL DOMAINS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 163
9.2 INVARIANT FACTORS OF A MATRIX . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 167
9.3 SIMILARITY INVARIANTS AND JORDAN REDUCTION . . . . . . . . . . . . .
. . . . . . . . 170
10 EXPONENTIAL OF A MATRIX, POLAR DECOMPOSITION, AND CLASSICAL GROUPS
183 10.1 THE POLAR DECOMPOSITION . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 183
10.2 EXPONENTIAL OF A MATRIX . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 184
10.3 STRUCTURE OF CLASSICAL GROUPS . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 188
10.4 THE GROUPS U ( P , Q ) . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 191
10.5 THE ORTHOGONAL GROUPS O ( P , Q ) . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 192
10.6 THE SYMPLECTIC GROUP SP N . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 195
IMAGE 3
CONTENTS IX
11 MATRIX FACTORIZATIONS AND THEIR APPLICATIONS . . . . . . . . . . . .
. . . . . . . . . 207
11.1 THE LU FACTORIZATION . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 208
11.2 CHOLESKI FACTORIZATION . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 213
11.3 THE QR FACTORIZATION . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 214
11.4 SINGULAR VALUE DECOMPOSITION . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 216
11.5 THE MOORE-PENROSE GENERALIZED INVERSE . . . . . . . . . . . . . . .
. . . . . . . . 218
12 ITERATIVE METHODS FOR LINEAR SYSTEMS . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 225
12.1 A CONVERGENCE CRITERION . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 226
12.2 BASIC METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 227
12.3 TWO CASES OF CONVERGENCE . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 229
12.4 THE TRIDIAGONAL CASE . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 231
12.5 THE METHOD OF THE CONJUGATE GRADIENT . . . . . . . . . . . . . . .
. . . . . . . . . . 235
13 APPROXIMATION OF EIGENVALUES . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 247
13.1 GENERAL CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 247
13.2 HESSENBERG MATRICES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 249
13.3 THE QR METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 253
13.4 THE JACOBI METHOD . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 259
13.5 THE POWER METHODS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 266
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 277
INDEX OF NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 279
GENERAL INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 283
CITED NAMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 289
|
| any_adam_object | 1 |
| author | Serre, Denis 1954- |
| author_GND | (DE-588)173306659 |
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| author_sort | Serre, Denis 1954- |
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| callnumber-first | Q - Science |
| callnumber-label | QA188 |
| callnumber-raw | QA188.S4713 2002 |
| callnumber-search | QA188.S4713 2002 |
| callnumber-sort | QA 3188 S4713 42002 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 140 SK 220 |
| classification_tum | MAT 150f |
| ctrlnum | (OCoLC)706019657 (DE-599)BVBBV036807374 |
| dewey-full | 512.9/43421 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 21 |
| dewey-search | 512.9/434 21 |
| dewey-sort | 3512.9 3434 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV036807374 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:48:40Z |
| institution | BVB |
| isbn | 9781441976826 9781441976833 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020723434 |
| oclc_num | 706019657 |
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| owner_facet | DE-706 DE-91G DE-BY-TUM DE-20 DE-11 DE-188 |
| physical | XIV, 289 S. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Serre, Denis 1954- Verfasser (DE-588)173306659 aut Matrices Matrices theory and applications Denis Serre 2. ed. New York [u.a.] Springer 2010 XIV, 289 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 216 Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s DE-604 Graduate texts in mathematics 216 (DE-604)BV000000067 216 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020723434&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Serre, Denis 1954- Matrices theory and applications Graduate texts in mathematics Matrices Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4037968-1 |
| title | Matrices theory and applications |
| title_alt | Matrices |
| title_auth | Matrices theory and applications |
| title_exact_search | Matrices theory and applications |
| title_full | Matrices theory and applications Denis Serre |
| title_fullStr | Matrices theory and applications Denis Serre |
| title_full_unstemmed | Matrices theory and applications Denis Serre |
| title_short | Matrices |
| title_sort | matrices theory and applications |
| title_sub | theory and applications |
| topic | Matrices Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrices Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020723434&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT serredenis matrices AT serredenis matricestheoryandapplications |