Verfügbarkeit: Matrices | Technische Universität München

Matrices: theory and applications

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Bibliographische Detailangaben
Beteilige Person:	Serre, Denis 1954- (VerfasserIn)
Format:	Buch
Sprache:	Englisch Französisch
Veröffentlicht:	New York [u.a.] Springer 2010
Ausgabe:	2. ed.
Schriftenreihe:	Graduate texts in mathematics 216
Schlagwörter:	Matrices Matrix > Mathematik
Links:	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
Umfang:	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
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spellingShingle	Serre, Denis 1954- Matrices theory and applications Graduate texts in mathematics Matrices Matrix Mathematik (DE-588)4037968-1 gnd
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title	Matrices theory and applications
title_alt	Matrices
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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
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