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Bhatia R. — Matrix Analysis
Bhatia R. — Matrix Analysis



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Название: Matrix Analysis

Автор: Bhatia R.

Аннотация:

The aim of this book is to present a substantial part of matrix analysis that is functional analytic in spirit. Much of this will be of interest to graduate students and research workers in operator theory, operator algebras, mathematical physics and numerical analysis. The book can be used as a basic text for graduate courses on advanced linear algebra and matrix analysis. It can also be used as supplementary text for courses in operator theory and numerical analysis. Among topics covered are the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, perturbation of matrix functions and matrix inequalities. Much of this is presented for the first time in a unified way in a textbook. The reader will learn several powerful methods and techniques of wide applicability, and see connections with other areas of mathematics. A large selection of matrix inequalities will make this book a valuable reference for students and researchers who are working in numerical analysis, mathematical physics and operator theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 347

Добавлена в каталог: 10.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L_{p}$-norms      84 89
$\nu$-mcasure of nonnormality      246
$\sin \Theta$ theorem      224
$\tau$-length of a path      175
$\tau$-optimal matching distance      173
Absolute value of a matrix      296
Absolute value of a vector      30
Absolute value, perturbation of      296
Adjoint      4
Analytic continuation      134
Analytic functions in a half-plane      134
Ando's Concavity Theorem      273
Angle operator      221
Angles between subspaces      201
Annihilation operator      21
Antisymmetric tensor power      16
Antisymmetric tensor product      16
Araki — Licb — Thirring inequality      258
Arithmetic-geometric mean inequality      87 262 295
Aronszajn's Inequality      64
Averaging      40 117
Bauer — Fike theorem      233
Bernstein's theorem      148
Bilinear functional      12
Bilinear functional, elementary      12
Bilinear map      12 310 313
Binomial operator      284
Biorthogonal      2
Birkhoff's theorem      37 180
Block-matrix      9 195
Borel measure      139
Bound norm      7
C-numerical radius      102 106
Canonical angles      201
Caratheodory's theorem      38
Carrollian n-tuple      242
Cartesian decomposition      6 25 73 156 183
Cauchy — Riemann equations      135
Cauchy — Schwarz inequality for matrices      266 286 297 298
Cauchy — Schwarz inequality for symmetric gauge functions      88
Cauchy — Schwarz inequality for unitarily invariant norms      95 108
Cauchy's integral formula      205
Cauchy's Interlacing Theorem      59
Cayley transform      284
Centraliser      167
Chain rule      311
Chebyshev's theorem      228
Cholesky decomposition      5 319
Class $\mathcal{L}$      268
Class $\mathcal{T}$      259
Coefficients in the characteristic polynomial      269
Commutant      167
Commutator      167 234
Compatible matching      36
Complete symmetric polynomial      18
Completely monotone function      148
Compression      59 61 119
Concave function      41 53 158 240 248 289
Condition number      232
Conformally equivalent      137
Contraction      7 119
Contractive      7 10
Convex function      40 41 45 87 117 157 218 240 248 265 281
Convex function, monotone      40
Convex function, smoothness properties      145
Convolution      146
Cosines of      201
Creation operator      21
CS decomposition      196 223
Cyclic order      184 191
Davis — Kahan $sin\Theta$ Theorem      212
Derivative      124 310
Determinant      16 253 269
Determinant, inequality      3 19 21 47 51 108 181 182 183 271 281
Determinant, perturbation of      22
Diagonal of A      37
Diagonalisable matrix      232
Diagonally dominant      251
Differentiable curve      166
Differentiable manifold      305
Differentiable map      124 310
Differentiation, rules of      311
Dilation Theory      26
Direct sum      9
Directional derivative      310
Distance between subspaces      202
Distribution function      139
Doubly stochastic matrix      23 32 165
Doubly substochastic matrix      38 39
Dual norm      89 96
Dual space      14
Dyson's expansion      311 319
Eigenvalues      24 58 67 77
Eigenvalues of A with respect to B      238
Eigenvalues of Hermitian matrices      24 35 62 101
Eigenvalues of product of unitary matrices      82
Eigenvalues, continuity of      152 168
Eigenvalues, continuous parametrization of      154
Eigenvalues, generalised      238
Elementary symmetric polynomi¬als      18 46 51
entropy      44 274 287
Entropy in quantum mechanics      274
Entropy, relative      274
Euclidean norm      86
Exponential      8 254 311
Extremal representation for singular values      75 76
Fan Dominance Theorem      93 284 291
Fan — Hoffman theorem      73
Field of values      8
First divided difference      123
Fischer's inequality      51
Fourier analysis      135
Fourier transform      206 207 216
Fourier transform, inverse      216
Fourier transform, minimal extrapolation problem      224
Frechet derivative      310
Frechet differentiable map      124
Frechet differential calculus      301 310
Frobenius norm      7 25 92 214
Fubini's theorem      140
Fuglede — Putnam theorem      235
Function of bounded variation      217
G Marriage Theorem      162 213
Gamma function      319
Gap      225
Gel'fand Nainiark Theorem      71
General linear group      7
Gersgorin Disk Theorem      244
Gersgorin disks      244
Golden — Thompson inequality      261 279 285
Gram determinant      20
Gram matrix      20
Gram — Schmidt procedure      2 287
Grassmann power      18
Hadamard Determinant Theo¬rem      24 46 51
Hadamard product      23
Hadamard's inequality      3 227
Hall's Marriage Theorem      36
Hall's theorem      52
Harmonic conjugates      137
Harmonic function      135
Harmonic function, harmonic function, mean value property      136
Hausdorff distance      160
Heinz inequalities      285
Henrici's Theorem      246
Herglotz theorem      136
Hermitian approxirnant      276
Hermitian matrix      4 8 57 155 254
Hilbert — Schmidt norm      7 92 97 299
Hoelder inequality      88
Hoelder inequality for symmetric gauge functions      88
Hoelder inequality for unitarily invariant norms      95
Hoffman — Wielandt inequality      165 237
Hoffman — Wielandt Theorem      165
Hyperbolic PDE      251
Hyperbolic polynomials      251
Imaginary part of a matrix      6
Inner product      1
Inner product, on matrices      92
Interlacing theorem      60
Interlacing theorem for singular values      81
Interlacing theorem, converse      61
Invariant subspace      10
Inverse      293
Isometry      119
Isotone      41
Jointly concave      271 273
Jordan canonical form      246
Jordan decomposition      99 262 277 292
Kato — Temple inequality      77
Koenig — Frobenius theorem      37
Krein — Milman theorem      133
Ky Fan k-norms      35 92
Ky Fan k-norms, main use      93
Ky Fan's Maximum Principle      24 35 65 69 71 174
Lattice superadditive      48 53
Laxly positive      238
Lebosguo Dominated Convergence Theorem      139
Length of a curve      169
Lexicographic ordering      14
Lidskii's theorem      69 179
Lidskii's Theorem for normal matrices      181
Lidskii's Theorem, multiplicative      73
Lidskii's Theorem, second proof      70
Lidskii's Theorem, third proof      98
Lie algebra      241
Lie bracket      167
Lie product formula      254 280
Lieb — Thirring inequality      279
Lieb's Concavity Theorem      271
Lieb's theorem      270
Liebian function      286
Lindblad's theorem      275
Lipschitz constant      215
Lipschitz continuous      322
Lipschitz continuous function      214
Loewdin orthogorialisation      287
Loewner — Heinz Theorem      285
Loewner's Theorems      131 149
Logarithm      145
Logarithm, principal branch      143
Logarithmic majorisation      71
Loowner — Heiuz inequality      150
LR decomposition      319
LR decomposition, pert urbation of      320
Lyapunov equation      221
Majorisation      28
Majorisation of complex vectors      179
Majorisation, soft      180
Majorisation, weak      30
Manifold      167
Marcus-de Oliviera Conjecture      184
Marriage problem      3
Matching problem      36
Matching theorem      185
Matrix convex      113
Matrix convex of order n      113
Matrix monotone      112
Matrix monotone of onlor      11 112
Matrix tiangle inequality      81
Matrix Young inequalities      286
Mean value theorem      303 307 312
Measure of nonnormality      245 251 278
Metrically flat      170
Mid-point operator confex      113
Minimax principle      58
Minimax principle for singular values      75
Minimax principle of Courant, Fischer and Weyl      58
Minkowski determinant inequality      56
Minkowski determinant theorem      47 282
Minkowski inequality, for symmetric gauge functions      89
Mirman's Theorem      25
Mixed Schwarz inequality      281 286
Mollifiers      146
Monotone      45 48
Monotone decreasing      41
Monotone increasing      41
Multi-indices      16
Multilinear functional      12
Multilinear map      12
Multiplication operator      223
Multiplication operator, left      273
Multiplication operator, right      273
Nearly normal matrix      252
Neumann series      7
Nevanlinna's theorem      138
Nevanlinna's Theorem, absolute      85
Nevanlinna's Theorem, bound      91
Nevanlinna's Theorem, Frobenius      92
Nevanlinna's Theorem, gauge invariant      85
Nevanlinna's Theorem, Hilbert-Schinidt      92
Nevanlinna's Theorem, Ky Fan      92
Nevanlinna's Theorem, monotone      85
Nevanlinna's Theorem, norm      1 6
Nevanlinna's Theorem, operator      91
Nevanlinna's Theorem, permutation invariant      85
Nevanlinna's Theorem, Schatten      92
Nevanlinna's Theorem, submultiplicative      94
Nevanlinna's Theorem, symmetric      85
Nevanlinna's Theorem, trace      92
Nevanlinna's Theorem, unitarily invariant      91
Nevanlinna's Theorem, weakly unitarily invariant      102
Normal approximant      277
Normal curve      169
Normal matrices, distance between eigenvalues      212
Normal matrices, path connected      169
Normal matrix      4 8 160 161 168 172 177 180 253
Normal matrix, function of      5
Normal matrix, spectral resolution      161
Normal path      169 177
Normal path inequality      189
Numerical radius      8 102
Numerical range      8 25
Operator approximation      192 275 287
Operator concave function      113 121
Operator convex function      113 130
Operator convex function, integral representation.      134
Operator monotone function      112 121 126 127 130 289 302 303 304 317
Operator monotone function, canonical representation      145
Operator monotone function, infinitely differentiable      134 290
Operator monotone function, integral representation      134
Operator monotone function, inverse of      293
Operator norm      7
Optimal matching      159
Optimal matching distance      52 153 160 21
Orbit      189
Orthostochastic matrix      35 180
P-Carrollian      242
p-norms      84
Partitioned matrix      64 188
Peierls — Bogoliubov inequality      275 281
Permanent      17 19
Permanent, inequality      19 21 23
Permanent, perturbation of      22
Permutation invariant      43
Permutation matrix      32 37 165
Permutation matrix, complex      85
Permutation orbit      166
permutations      165
Pick functions      135 139
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