Stewart G.W., Sun J. — Matrix perturbation theory :: Электронная библиотека попечительского совета мехмата МГУ

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Stewart G.W., Sun J. — Matrix perturbation theory

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Название: Matrix perturbation theory

Авторы: Stewart G.W., Sun J.

Аннотация:

This book is a comprehensive survey of matrix perturbation theory, a topic of interest to numerical analysts, statisticians, physical scientists, and engineers. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the generalized eignevalue problem as wellas a complete treatment of vector and matrix norms, including the theory of unitary invariant norms.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 363

Добавлена в каталог: 09.12.2005

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Предметный указатель

Idempotent matrix      28
Inertia of a matrix      196
Inner product      53 62
Invariant subspace      21 22
Invariant subspace, approximate      (see Invariant subspace residual
Invariant subspace, backward perturbation      175 178—180
Invariant subspace, backward perturbation, optimal      176
Invariant subspace, characterization      220 227
Invariant subspace, complementary      222 225
Invariant subspace, complex conjugate, eigenvalues      29
Invariant subspace, definition      22
Invariant subspace, left      221 225
Invariant subspace, normalization      239 244
Invariant subspace, perturbation theory      229 236 240 244 254
Invariant subspace, perturbation theory, $tan \Theta$ theorem      253 258 259
Invariant subspace, perturbation theory, canonical angles      237
Invariant subspace, perturbation theory, first $sin \Theta$ theorem      250 258
Invariant subspace, perturbation theory, Hermitian matrix      244
Invariant subspace, perturbation theory, second $sin \Theta$ theorem      251 255 258
Invariant subspace, reduced form of matrix      221
Invariant subspace, reduction to block diagonal form      224
Invariant subspace, representation of matrix      22 220 231 235 237
Invariant subspace, representation of matrix, condition number      240
Invariant subspace, residual bound      174 206 229—236 246 249 254
Invariant subspace, residual bound, canonical angles      232
Invariant subspace, residual bound, nonorthonorinal basis      251
Invariant subspace, sensitivity      21 90 219
Invariant subspace, simple      220 221 226 227 231 235 238
Invariant subspace, simple, exist ence of complementary invariant subspace      225
Invariant subspace, spectral projection      114 225 240
Invariant subspace, spectral projection, canonical angles      226
Invariant subspace, spectral resolution      223 228 237 244
Invariant subspace, Sylvester’s equation (q.v.)      222
Inverse matrix, artificial ill-conditioning      122 124
Inverse matrix, asymptotic forms and derivatives      130—132
Inverse matrix, condition      17
Inverse matrix, condition number      119 127 133
Inverse matrix, condition number inverse matrix      102
Inverse matrix, condition number, 2—norm      121
Inverse matrix, condition number, optimal      133 135 193
Inverse matrix, condition number, relation to determinant      134
Inverse matrix, condition number, relation to eigenvalues      122 134 135
Inverse matrix, condition number, significant digits in inverse      120
Inverse matrix, distance from singularity      120 121 133
Inverse matrix, left and right inverses      134
Inverse matrix, perturbation theory      117—124
Inverse matrix, perturbation theory, linear system      124
Inverse matrix, perturbation theory, random perturbations      131
Inverse matrix, well-conditioned      120
Irreducible matrix      186 188
Jacobi, C. G. J.      209
Jacobian matrix      134
Jiang, E.      178 180
Jordan block      20 28 174 180 186 300
Jordan block, function of      73
Jordan block, powers of      28
Jordan canonical form      20 21 26 174 227 280
Jordan canonical form, associated invariant subspaces      21
Jordan canonical form, computation      27
Jordan canonical form, Drazin generalized inverse      113
Jordan canonical form, limitations      21 227
Jordan canonical form, principal vector      21
Jordan Wielandt matrix      32 34 35 259
Jordan, C.      26 34 35 45 289
Jordan, P.      63
Kalian, W.      27 45 46 133 151 178 180 209 211 217 218 244 258 259
Kato, T.      4 176 178 244
Konig, D.      88
Korganoff, A.      152
Krasnoselski, M. A.      98
Krein, M. G.      98
Kronecker, L.      289
Kroticcker product      30 228 258
Lancaster, P.      227
Laplace, P. S.      11
Lawson, C. L.      152 163
Least squares      10 11 101
Least squares, asymptotic forms and derivatives      162
Least squares, backward perturbation      160—163
Least squares, backward perturbation, Bjoerck’s theorem      162
Least squares, condition, condition number      156 163
Least squares, condition, reflected by solution      156 158 163
Least squares, condition, square of $\kappa$       158 163
Least squares, constrained      109 112
Least squares, cross-product matrix backward perturbation      161
Least squares, elliptic norm      109
Least squares, errors in the variables      163
Least squares, errors in the variables, bias      267
Least squares, expanded equations      107 161 163
Least squares, Gauss — Markov theorem      110
Least squares, Gauss. C. F.      108
Least squares, measurement error models      163
Least squares, normal equations      107
Least squares, numerical methods      152 163
Least squares, perturbation theory      156 160
Least squares, perturbation theory, Bjoerck’s theorem      158
Least squares, perturbation theory, errors in a column      163
Least squares, perturbation theory, perturbation in A      157
Least squares, perturbation theory, perturbation in b      156
Least squares, perturbation theory, perturbation of the residual vector      160
Least squares, perturbation theory, structured perturbation      158 163
Least squares, priority dispute between Gauss and Legendre      109
Least squares, reduced form      155
Least squares, regression diagnostics      163
Least squares, residual vector      107
Least squares, solution by pseudo-inverse      107
Least squares, statisticians’ notation      109
Legendre, A. M.      109
Levy, L.      186
Lidskii, V. D.      209 210
Linear functional      56
Linear system      101 114
Linear system, artificial ill-conditioning      128
Linear system, asymptotic forms and derivatives      130—132
Linear system, backward perturbation      128—130 133 135 136
Linear system, Bauer — Skeel      128 133
Linear system, Bauer — Skeel theorem      127
Linear system, component-wise bounds      125 126
Linear system, condition number      125 127
Linear system, Oettli Prager theorem      130
Linear system, perturbation theory      124 128 132
Linear system, perturbation theory from inverse matrix      124
Linear system, perturbation theory, perturbation in matrix      124
Linear system, perturbation theory, perturbation in the right-hand side      126 127
Linear system, perturbation theory, structured perturbation      127 128
Linear system, reflected by solution      126
Linear system, residual bound      128 130
Linear system, Rigal Cinches theorem      128
Linear system, structured      129
Linear system, structured perturbation      133
LINPACK      133
Littlewood, J. E.      88
Loewy, A.      27
LR algorithm      11
Lu, Q.-k      99
Mac Duffee, C. C.      4
Majorization      81 88 89
Marcus, M.      4
Matching distance      (see Under eigenvalue) 167
Matrix function      20
Matrix function, exponential      73
Matrix function, Jordan block      73
Matrix function, Neumann series      73
Matrix function, power series      73
Matrix function, rational      29
Matrix function, spectral resolution      229
Matrix norm      (see under norm) 64
Matrix pair (definite)      281—283 289
Matrix pair (definite), $sin \Theta$ theorems      323 324
Matrix pair (definite), bounds for dif      319

Matrix pair (definite), condition number      313 316 324
Matrix pair (definite), definition      282
Matrix pair (definite), diagonalizability      283
Matrix pair (definite), eigenangle      314
Matrix pair (definite), eigenangle, perturbation theory      324
Matrix pair (definite), eigenangle, variational characterization      314
Matrix pair (definite), eigenangles      324
Matrix pair (definite), eigenvector      318
Matrix pair (definite), failure of definition in real case      290
Matrix pair (definite), Fischer’s theorem      281
Matrix pair (definite), limitations of the theory      316
Matrix pair (definite), matching distance      324
Matrix pair (definite), perturbation of eigenspaces      317 322
Matrix pair (definite), perturbation of eigenvalues      314 317
Matrix pair (definite), perturbation of eigenvectors      322
Matrix pair (definite), positive definite B      281 282 289
Matrix pair (definite), projective metric      285—287 290 314
Matrix pair (definite), right and left eigenspaces      317
Matrix pair (definite), spectral resolution      318
Matrix pair (definite), spectral resolution normalized      318
Matrix pair (regular)      273 274
Matrix pair (regular), approximate eigenspace      306—309
Matrix pair (regular), condition number      294 300 303
Matrix pair (regular), continuity of eigenvalues      291
Matrix pair (regular), deflating subspace      311
Matrix pair (regular), diagonalizable pair      300 303
Matrix pair (regular), eigenspace      312
Matrix pair (regular), eigenspace, backward perturbation      309
Matrix pair (regular), eigenspace, characterizations      304
Matrix pair (regular), eigenspace, complementary      306
Matrix pair (regular), eigenspace, definition      303
Matrix pair (regular), eigenspace, eigenvectors      312
Matrix pair (regular), eigenspace, simple      277
Matrix pair (regular), first order approximation      292
Matrix pair (regular), first order error bounds      293
Matrix pair (regular), generalized Bauer Fike theorem      301
Matrix pair (regular), generalized Bauer-Fike, theorem      294 301 311
Matrix pair (regular), generalized Henrici Theorem      311
Matrix pair (regular), generalized Sehur form      276 289
Matrix pair (regular), generalized Sehur form, real pair      290
Matrix pair (regular), generalized, Hoffman Wielandt theorem      311
Matrix pair (regular), Gerschgorin theory      294 300
Matrix pair (regular), Gerschgorin theory, diagonal similarity      297
Matrix pair (regular), Gerschgorin theory, generalized Gerschgorin theorem      295
Matrix pair (regular), Gerschgorin theory, simplification of bounds      296
Matrix pair (regular), infinite eigenvalues      274
Matrix pair (regular), loft projective metric      288 290
Matrix pair (regular), normal pair      311
Matrix pair (regular), number of eigenvalues      275
Matrix pair (regular), perturbation of eigenspaces      309—311
Matrix pair (regular), perturbation of eigenvalues      311
Matrix pair, characteristic, equation      274
Matrix pair, eigenvalue      273
Matrix pair, eigenvalue problem      274
Matrix pair, eigenvalue, $(\alpha/\beta)$ notation      273
Matrix pair, eigenvector      273
Matrix pair, equivalence      276
Matrix pair, generalized Schur form      277
Matrix pair, Hermitian      281
Matrix pair, infinite eigenvalues      271 273
Matrix pair, limitations      288
Matrix pair, linear transformations of matrix pairs      275
Matrix pair, metrics      284 288
Matrix pair, nonsingular B      274
Matrix pair, numerical methods      274
Matrix pair, scaling problems      272
Matrix pair, singular      273 274 289
Matrix pair, systems of differential equations      289
Matrix pencil      209 271
Matrix pencil, rectangular      272
Matrix pencil, singular      271
Mcintosh, A.      194 227 228
Metric      53
Metric from a norm      62
Metric, pseudo-metric      62
Metrics for subspaces      50 90
Metrics for subspaces, $\theta$ -metric      96 98 99
Metrics for subspaces, basis metric      95 98 99
Metrics for subspaces, completeness      99
Metrics for subspaces, distance from a vector to a subspace      90 99
Metrics for subspaces, failure to generate the gap topology      94
Metrics for subspaces, gap function      90 94 98 99
Metrics for subspaces, gap function, canonical angles      92
Metrics for subspaces, gap function, definition      91
Metrics for subspaces, gap function, equivalence of gap functions      91
Metrics for subspaces, gap function, equivalence of gap, projections      93
Metrics for subspaces, gap topology      91 93 94
Metrics for subspaces, projection metrics      93 94
Metrics for subspaces, Schaffer’s metric      99
Metrics for subspaces, unitarily invariant, metrics      94—98
Metrics for subspaces, unitary invariance      99
Meyer, C.      244
Milman, D. P.      98
Mine, H.      4
Minkowski, H.      59 61
Minkowski’s inequality      61
Mirsky, L.      71 72 194 209 210
Mirsky’s theorem      194 204 206 208 209
Moler, C. B.      290
Moore, E. H.      108 109
Moore-Penrose generalized inverse      (see pseudo-inverse) 102
More, G.      135
Multiple eigenvalue      297 300
Multiple eigenvalue, perturbation of eigenvectors      312
Multiple eigenvalue, recovery of bounds for ordinary eigenvalue problem      294 297
Multiple eigenvalue, right projective metric      288 290
Multiple eigenvalue, spectral resolution      300 310
Multiple eigenvalue, spectral variation      302
Multiple eigenvalue, Weierstrass canonical form      280 290
Nashed, M. Z.      108
Nearly singular matrix      120
Neumann series      73
Nilpotent matrix      29
Nondefective matrix      21 28
Norm and spectral radius      73
Norm of a linear functional      56
Norm, 2-norm (q.v.)      3
Norm, absolute norm      52 72 76
Norm, combining norms      61
Norm, consistency      65 67 71
Norm, consistency, definition      66
Norm, consistency, failure      65
Norm, consistency, family of consistent norms      69
Norm, consistency, unitarily invariant norms      80
Norm, consistency, vector norm consistent with a matrix norm      66
Norm, convexity and norms      59 63
Norm, dual      57 59 67
Norm, dual of dual      58
Norm, elementary properties      51
Norm, elliptic      53 62 111
Norm, equivalence of norms      54 55 59 65 72
Norm, failure      64
Norm, Frobenius norm (q.v.)      65
Norm, generated by an inner product      62
Norm, generated by linear transformation      53
Norm, generated by positive definite matrix      53 62
Norm, Hilbert — Schmidt norm      210
Norm, Hoelder norms      51 57 60 61 64 69 121 134
Norm, infinite dimensional spaces      71
Norm, limits and norms      55 65
Norm, matrix $\infty$ -norm      69 71 72
Norm, matrix 1—norm      69 71 72
Norm, matrix norm      49 64
Norm, operator norm      67 68 72
Norm, operator norm, consistency      67 68
Norm, polar      59
Norm, spectral radius and norms      67 72 73
Norm, subordinate norm      (see norm operator
Norm, unitarily invariant (q.v.)      50
Norm, vector $\infty$ -norm      51 55 69

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