Matrices: algebra, analysis, and applications
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2016]
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | xii, 582 Seiten |
| ISBN: | 9789814667968 9789813141032 |
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MARC
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| 245 | 1 | 0 | |a Matrices |b algebra, analysis, and applications |c Shmuel Friedland University of Illinois at Chicago, USA |
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Record in the Search Index
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| adam_text | ALGEBRA, ANALYSIS AND APPLICATIONS
This volume deals with advanced topics in matrix theory using
the notions and tools from algebra, analysis, geometry and
numerical analysis. It consists of seven chapters that are loosely
connected and interdependent. The choice of the topics is very
personal and reflects the subjects that the author was actively
working on in the last 40 years. Many results appear for the
first time in the volume. Readers will encounter various
properties of matrices with entries in integral domains,
canonical forms for similarity, and notions of analytic, pointwise
and rational similarity of matrices with entries which are locally
analytic functions in one variable. This volume is also devoted
to various properties of operators in inner product space, with
tensor products and other concepts in multilinear algebra, and
the theory of non-negative matrices. It will be of great use to
graduate students and researchers working in pure and applied
mathematics, bioinformatics, computer science, engineering,
operations research, physics and statistics.
Contents
Preface vii
1 Domains, Modules and Matrices 1
1.1 Rings, Domains and Fields ........................ 1
1.2 Bezout Domains .................................... 4
1.3 D[/,Dp and Oe Domains ........................... 9
1.4 Factorizations in D[r] ........................... 12
1.5 Elementary Divisor Domains ....................... 16
1.6 Modules........................................... 17
1.7 Determinants...................................... 25
1.8 Algebraically Closed Fields....................... 30
1.9 The Resultant and the Discriminant.................32
1.10 The Ring F[æi,... ,xn]............................ 37
1.11 Matrices and Homomorphisms ....................... 38
1.12 Hermite Normal Form............................... 42
1.13 Systems of Linear Equations over Bezout
Domains......................................... 51
1.14 Smith Normal Form............................... 55
1.15 Local Analytic Functions in One Variable.......... 60
1.16 The Local-Global Domains in ............ 67
1.17 Historical Remarks.............................. 74
2 Canonical Forms for Similarity Y5
2.1 Strict Equivalence of Pencils................... 75
2.2 Similarity of Matrices ........................... 82
2.3 The Companion Matrix.............................. 84
ix
x Contents
2.4 Splitting to Invariant Subspaces..................... 87
2.5 An Upper Triangular Form............................. 92
2.6 Jordan Canonical Form................................ 96
2.7 Some Applications of Jordan Canonical Form............99
2.8 The Matrix Equation AX — XB — 0 103
2.9 A Criterion for Similarity of Two Matrices...........109
2.10 The Matrix Equation AX — XB — C .....................114
2.11 A Case of Two Nilpotent Matrices.....................119
2.12 Historical Remarks................................. 122
3 Functions of Matrices and Analytic Similarity 123
3.1 Components of a Matrix and Functions
of Matrices..........................................123
3.2 Cesaro Convergence of Matrices.......................127
3.3 An Iteration Scheme..................................131
3.4 Cauchy Integral Formula for Functions
of Matrices..........................................133
3.5 A Canonical Form over ...............................143
3.6 Analytic, Pointwise and Rational Similarity..........150
3.7 A Global Splitting...................................154
3.8 First Variation of a Geometrically Simple
Eigenvalue...........................................155
3.9 Analytic Similarity over Ho..........................157
3.10 Strict Similarity of Matrix Polynomials..............166
3.11 Similarity to Diagonal Matrices......................169
3.12 Property L ..........................................179
3.13 Strict Similarity of Pencils and Analytic
Similarity...........................................185
3.14 Historical Remarks...................................192
4 Inner Product Spaces 195
4.1 Inner Product........................................195
4.2 Special Transformations in IPS..................... 200
4.3 Symmetric Bilinear and Hermitian Forms...............209
4.4 Max—Min Characterizations of Eigenvalues.............212
4.5 Positive Definite Operators and Matrices.............221
Contents xi
4.6 Convexity ........................................227
4.7 Majorization .....................................237
4.8 Spectral Functions.............................. 245
4.9 Inequalities for Traces...........................253
4.10 Singular Value Decomposition (SVD) ...............256
4.11 Characterizations of Singular Values..............263
4.12 Moore-Penrose Generalized Inverse.................273
4.13 Approximation by Low Rank Matrices................276
4.14 CUR-Approximations ...............................281
4.15 Some Special Maximal Spectral Problems............285
4.16 Multiplicity Index of a Subspace of S(V) . .......292
4.17 Rellich’s Theorem .............................. 300
4.18 Hermitian Pencils ..................... 304
4.19 Eigenvalues of Sum of Hermitian Matrices......... . 316
4.20 Perturbation Formulas for Eigenvalues
and Eigenvectors of Hermitian Pencils.............319
4.21 Historical Remarks................................323
5 Elements of Multilinear Algebra 325
5.1 Tensor Product of Two Free Modules................325
5.2 Tensor Product of Several Free Modules............333
5.3 Sparse Bases of Subspaces.........................344
5.4 Tensor Products of Inner Product Spaces...........360
5.5 Matrix Exponents .................................374
5.6 Historical Remarks................................392
6 Non-Negative Matrices 393
6.1 Graphs.......................................... 393
6.1.1 Undirected graphs..........................393
6.1.2 Directed graphs ...........................394
6.1.3 Multigraphs and multidigraphs............ 395
6.1.4 Matrices and graphs...................... 399
6.2 Perron—Frobenius Theorem ............... 403
6.3 Index of Primitivity ........................... 416
6.4 Reducible Matrices................................418
6.5 Stochastic Matrices and Markov Chains.............427
xii Contents
6.6 Friedland-Karlin Results............................437
6.7 Log-Convexity ......................................447
6.8 Min-Max Characterizations of ) ...............450
6.9 An Application to Cellular Communication............456
6.9.1 Introduction ................................456
6.9.2 Statement of problems........................457
6.9.3 Relaxations of optimal problems..............459
6.9.4 Preliminary results..........................460
6.9.5 Reformulation of optimal problems............464
6.9.6 Algorithms for sum rate maximization.........467
6.10 Historical Remarks..................................472
7 Various Topics 473
7.1 Norms over Vector Spaces............................473
7.2 Numerical Ranges and Radii .........................482
7.3 Superstable Norms...................................490
7.4 Operator Norms .....................................494
7.5 Tensor Products of Convex Sets......................498
7.6 The Complexity of conv Cln © f ...........508
7.7 Variation of Tensor Powers and Spectra..............511
7.8 Variation of Permanents.............................521
7.9 Vivanţi—Pringsheim Theorem and Applications .... 525
7.10 Inverse Eigenvalue Problem for Non-Negative
Matrices............................................533
7.11 Cones...............................................542
7.12 Historical Remarks..................................554
Bibliography 555
Index of Symbols 571
Index 579
|
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| classification_rvk | SK 220 |
| classification_tum | MAT 150f |
| contents | Includes bibliographical references and index |
| ctrlnum | (OCoLC)950255232 (DE-599)GBV826142052 |
| discipline | Mathematik |
| format | Book |
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| isbn | 9789814667968 9789813141032 |
| language | English |
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| spelling | Friedland, Shmuel 1944- Verfasser (DE-588)113767021 aut Matrices algebra, analysis, and applications Shmuel Friedland University of Illinois at Chicago, USA New Jersey World Scientific [2016] © 2016 xii, 582 Seiten txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Matrices Eigenvalues Non-negative matrices Eigenwert (DE-588)4151200-5 gnd rswk-swf Nichtnegative Matrix (DE-588)4310434-4 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s Eigenwert (DE-588)4151200-5 s Nichtnegative Matrix (DE-588)4310434-4 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028326387&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028326387&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Friedland, Shmuel 1944- Matrices algebra, analysis, and applications Includes bibliographical references and index Matrices Eigenvalues Non-negative matrices Eigenwert (DE-588)4151200-5 gnd Nichtnegative Matrix (DE-588)4310434-4 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4151200-5 (DE-588)4310434-4 (DE-588)4037968-1 |
| title | Matrices algebra, analysis, and applications |
| title_auth | Matrices algebra, analysis, and applications |
| title_exact_search | Matrices algebra, analysis, and applications |
| title_full | Matrices algebra, analysis, and applications Shmuel Friedland University of Illinois at Chicago, USA |
| title_fullStr | Matrices algebra, analysis, and applications Shmuel Friedland University of Illinois at Chicago, USA |
| title_full_unstemmed | Matrices algebra, analysis, and applications Shmuel Friedland University of Illinois at Chicago, USA |
| title_short | Matrices |
| title_sort | matrices algebra analysis and applications |
| title_sub | algebra, analysis, and applications |
| topic | Matrices Eigenvalues Non-negative matrices Eigenwert (DE-588)4151200-5 gnd Nichtnegative Matrix (DE-588)4310434-4 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrices Eigenvalues Non-negative matrices Eigenwert Nichtnegative Matrix Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028326387&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028326387&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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