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

$in_{2}$ in terms of the inverse matrix      36
$in_{2}$ in terms of the pseudo-inverse      110
2—norm      51 59 71 72
2—norm as largest singular value      33 69
2—norm, consistency with Frobenius norm      66
2—norm, matrix 2—norm      69
2—norm, properties      36 51 70
2—norm, relation to the 1—norm and $\infty$ -norin      55
2—norm, symmetric gauge function      79
2—norm, unitary invariance      52 60 72 74
2—norm, vector 2—norm      3
Abdelmalek, N. N.      163
Absolute and relative error in individual elements      128
Absolute and relative error, matrix      116 117 119 134
Absolute and relative error, matrix, limitations      117
Absolute and relative error, matrix, properties      116
Absolute and relative error, scalar      115
Absolute and relative error, scalar, properties      115
Absolute value      49
Acute perturbation      136 140 151 152
Acute perturbation, continutiy of pseudo-inverse      140
Acute perturbation, definition      139
Acute perturbation, in reduced form      139
Acute subspaces      151 152 255
Afrial, S. N.      45
Amir — Moez, A. R.      209
Approximation by matrix of lower rank      (see Sclnnidt Mirsky theorem 208)
Ariolo, M.      163
Arithmetic-geometric mean, inequality      61
Artificial ill-conditioning      125 193
Autoime, L.      35 36
Backward perturbation      (see under linear system least etc
Banach space      60 98
Banach, S.      60
Bateman, H.      35
Bauer Fike theorem      171 181 192 294 300
Bauer Skeel theorem      127
Bauer, F. L.      133 177 194 195
Baumgaertel, II.      176
Beckenbach, E. F.      60
Bellman, R.      4 60
Beltrami, E.      34
Ben — Israel, A      152
Bendixson Hirsch Toeplitz, theoroni      30
Bendixson, I.      30
Bergman, P. G.      108
Berkson, E.      98
Bhatia, R.      17
Birkhoff, G. D.      88
Birkhoff’s theorem      85
Bjerhairmier, A.      108 109
Bjoerck, A.      163
Block      4
Borchatt, C. W.      209
Bunch, J. R.      136
Bunyakovski      60
Canonical angle      43 45 94 98 99 226 232 240 250 255 260 323
Canonical angle, basis metric      95
Canonical angle, computation      43 45
Canonical angle, gap function      92
Canonical angle, pairs of projections      43
Canonical angle, variational characterization      45
Canonical bases      40
Cauchy inequality      5 60
Cauchy inequality, generalized      67
Cauchy sequence      63 64 99
Cauchy, A. L.      60 209
Cauchy’s interlacing theorem      196 198 209
Cayley Hamilton theorem      27
Characteristic equation      15
Characteristic polynomial      15
Chatelin, P.      4
Chebyshev, P. L.      11
Cholesky factor      13
Chordal metric      283 284 290 314
Column space      2 4
Column sum norm      (sec Norm matrix
Companion matrix      28
Complete space      63 64 99
Condition estimation      133
Condition number      (see Under matrix inverse eigenvalue etc.)
Congruence transformation      196
Consistency, between norms      (see norm consistency)
Consistency, foolish      26
Contract,ion matrix      46
Courant, R.      209
Courant-Fischer theorem      (see Fischer’s theorem) 209
Crawford, C. R.      290 324
CS decomposition      37 40 45
CS decomposition, computation      45
CS decomposition, existence      37
CS decomposition, generalized singular value decomposition      47
CS decomposition, perturbation theory      324
Davis, C.      45 46 151 194 227 228 244 258 259
Decell, H. P.      152
Defective matrix      16
Defective matrix, Jordan form      21
Definite matrix pair      (see Matrix pair (definite)) 281
Definition      81
Deminel, J.      133 136 177
Dennis, J. E.      135
Departure from normality      171 172 177 178
Desplanques, J.      186
Determinant as poor measure of condition      122
Diagonal matrix      3 5
Diagonalizable matrix      21 28
Diagonally dominant matrix      186—188
DIF      307 309 311 312
Dif, bounds for definite matrix, pair      319
Dilation      209 211
Direct rotation      46
Doubly stochastic matrix      83 85 88 195
Drazin, M. P.      108 113
Duff, I. S.      163
Dulrnage, L.      88
Eckart, C.      35 210
Eckart-Young theorem      (see Schmidt — Mirsky theorem) 210
Eigenpair      14
Eigenproblem      26
Eigenspace      (see Under matrix pair) 303
Eigensystein      26
Eigenvalue      14
Eigenvalue, algebraic multiplicity      15 16
Eigenvalue, backward perturbation      175
Eigenvalue, Cauchy’s interlacing theorem (q.v.)      196
Eigenvalue, complex      15
Eigenvalue, condition number      186 188 192 226 240
Eigenvalue, continuity      166 167 176 178 244
Eigenvalue, defective      16 176
Eigenvalue, differentiability      185
Eigenvalue, Fischer’s theorem (q.v.)      27
Eigenvalue, geometric multiplicity      15 16
Eigenvalue, Gerschgorin disks (q.v.)      181
Eigenvalue, Gerschgorin’s theorem (q.v.)      181
Eigenvalue, Hausdorff distance      167—169 177 178
Eigenvalue, inclusion region      195 210
Eigenvalue, matching distance      167 169 174 177 178 217
Eigenvalue, matching distance, $md_{2}$       189
Eigenvalue, matrix pair (q.v.)      271
Eigenvalue, multiple      15 26 27
Eigenvalue, nomenclature      14 26
Eigenvalue, nomenclature of matrix functions      29
Eigenvalue, optimal      176
Eigenvalue, perturbation theory      165 176 203 241
Eigenvalue, perturbation theory, Bauer-Fike theorem (q.v.)      171
Eigenvalue, perturbation theory, diagonalizable matrix      192
Eigenvalue, perturbation theory, Eisner’s theorem (q.v.)      168
Eigenvalue, perturbation theory, generalized Rayleigh quotient      249
Eigenvalue, perturbation theory, Henrici’s theorem (q.v.)      172
Eigenvalue, perturbation theory, Hermitian matrix      258 263

Eigenvalue, perturbation theory, Hoffman-Wielandt theorem (q.v.)      189
Eigenvalue, perturbation theory, matrices similar to a Hermitian matrix      215 216
Eigenvalue, perturbation theory, matrices similar to a normal matrix      216
Eigenvalue, perturbation theory, Mirsky’s theorem (q.v.)      205
Eigenvalue, perturbation theory, non-Hermitian perturbations of Hermitian matrices      212 214 217
Eigenvalue, perturbation theory, normal matrix      192 195
Eigenvalue, perturbation theory, orthogonal matrix      195
Eigenvalue, perturbation theory, Ostrowski-Elsner theorem (q.v.)      170
Eigenvalue, perturbation theory, simple eigenvalue      183
Eigenvalue, perturbation theory, Weyl’s theorem (q.v.)      203
Eigenvalue, Rayleigh quotient (q.v.)      185
Eigenvalue, residual bound      174 176 191 205 207 209 211 235 248 254 257
Eigenvalue, set of eigenvalues      26
Eigenvalue, simple      15 29 183
Eigenvalue, spectral variation      167 169 177
Eigenvector      14
Eigenvector, backward pertubation      179
Eigenvector, condition      219
Eigenvector, condition number      241
Eigenvector, left      15
Eigenvector, nonuniqueness      89 219 220
Eigenvector, perturbation theory      240 241 244
Eigenvector, perturbation theory, Hermitian matrix      258 263
Eigenvector, perturbation theory, stochastic matrix      241 244 246
Eigenvector, residual bound      211
Eigenvector, right      15
Eigenvector, simple eigenvalue      29
Eisner, L.      177 178 311
Eisner’s theorem      167 168 170 181
Elliptic norm      109
Error      (see absolute and relative error) 115
Euclidean norm      3 53
Evans, J. W.      152
Exponential of a matrix      73
Faddeeva, V. N.      71
Fan, K.      88
Fan’s theorem      50 81 86 254
Feingold, D. B.      188
Feller, W.      10
Field of values      23 24 27
Field of values of a Hermitian matrix      28
Field of values of a normal matrix      24
Field of values, convex hull of eigenvalues      24
Field of values, convexity of      23
Field of values, definition      23
Fike, C. T.      177 194
First order approximation      131 134 292 eigenvalue etc.)
Fischer, E.      209 289
Fischer’s theorem      27 196 198 201 209 281 289
Forsythe, G. E.      10
Francis G. F.      11
Frobenius norm      65 71 110 131 135 172 177 180 247 258
Frobenius norm, consistency      65 69 71
Frobenius norm, consistency with 2—norm      66
Frobenius norm, relation to eigenvalues      72
Frobenius norm, symmetric gauge function      79
Frobenius norm, unitary invariance      71 72 74
Frobenius, F. G.      71 88
Full rank factorization      12 32 105
Gaches, J.      134
Gantmacher, F. R.      4
Gap function      (see under metrics for subspaces) 90
Gastinel, N.      71 133
Gatlinburg Conferences      71
Gauss, C. F.      108 110 134
Gaussian elimination      132
Generalized      240 244 248 249
Generalized eigenvalue problem      (see matrix pair) 271
Generalized inverse      102
Generalized inverse, (i,j,k)-inverse      102
Generalized inverse, discontinuity      108
Generalized inverse, Drazin inverse      108 113 241
Generalized inverse, from singular value decomposition      103 104 110
Generalized inverse, group inverse      241
Generalized inverse, limitiations      108
Generalized inverse, projections      110
Generalized singular value decomposition      46
Generalized singular value decomposition, perturbation theory      324
Gerschgorin disks      181 186 187
Gerschgorin disks, irreducible matrix      188
Gerschgorin disks, isolated      187
Gerschgorin disks, similarity      182—187
Gerschgorin, S. A.      177 186
Gerschgorin’s theorem      177 181 186 187 203
Gerschgorin’s theorem, block variant      188
Gerschgorin’s theorem, compared with Eisner’s theorem      181
Gerschgorin’s theorem, generalized      (see Under matrix pair (regular)) 291
Gohberg, I.      227
Golub, G. H.      4 151 152 155 163
Gragg, W. B.      133
Gram — Schmidt algorithm      11 12
Gram — Schmidt algorithm, modified      12
Gram, J. P.      11
Hadamard’s inequality      8 14 168 177
Hahn Banach theorem      57 63
Hahn, H.      60
Hall, P.      88 89
Hall’s theorem      84 88 89 170 178
Halmos, P. R.      46
Halperin, I.      88
Hanson, R.      152 163
Hardy — Littlewood — Polya Theorem      81 87
Hardy, G. H.      88
Hausdorff distance      (see Under eigenvalue) 167
Hausdorff, F.      27 177
Hearon, J. Z.      152
Henna, P.      177 178
Henrici’s theorem      172 174 177
Hermite, C.      209
Hermitian matrix      3 5
Hermitian matrix, Cauchy’s interlacing theorem (q.v.)      196
Hermitian matrix, eigenvalues      19
Hermitian matrix, eigenvalues of sums      25
Hermitian matrix, eigenvalues, bounds      30
Hermitian matrix, field of values      28
Hermitian matrix, Fischer’s theorem (q.v.)      27
Hermitian matrix, nonorthonorinal baisis      251
Hermitian matrix, perturbation of eigenvales, Mirsky’s theorem (q.v.)      204
Hermitian matrix, perturbation of eigenvalues      203 258 263
Hermitian matrix, perturbation of eigenvalues, first $sin \Theta$ theorem      250 258
Hermitian matrix, perturbation of eigenvalues, generalized Rayleigh Quotient      249
Hermitian matrix, perturbation of eigenvalues, matrices similar to a Hermitian matrix      215 216
Hermitian matrix, perturbation of eigenvalues, non-Hermitian perturbations      212 214 217
Hermitian matrix, perturbation of eigenvalues, perturbation of eigenvectors      258 263
Hermitian matrix, perturbation of eigenvalues, perturbation of invariant subspaces      244
Hermitian matrix, perturbation of eigenvalues, positive definite perturbation      203
Hermitian matrix, perturbation of eigenvalues, residual bound for eigenvalues      205 207
Hermitian matrix, perturbation of eigenvalues, residual bounds for eigenvalues      248 254 257
Hermitian matrix, perturbation of eigenvalues, residual bounds for invariant subspac.es      249 254
Hermitian matrix, perturbation of eigenvalues, Weyl’s theorem (q.v.)      203
Hermitian matrix, second $sin \Theta$ theorem      251 255 258
Hermitian matrix, skew Hermitian matrix      5
Hermitian matrix, spectral decomposition      19 226
Hermitian matrix, sums of      27
Hermitian matrix, tan $sin \Theta$ theorem      253 258 259
Hessian matrix      134
Higham, N. J.      133 164
Hilbert space      64 98
Hirsh, H.      30
Hoelder norms      192
Hoelder, O.      61
Hoelder’s inequality      61
Hoffman Wielandt, generalizations      193 194
Hoffman Wielandt, limitations      191
Hoffman Wielandt, theorem      189 193 205 213 218 257
Hoffman, A. J.      88 193
Holbrook, J. R. A.      194
Hotelling, H.      35
Householder transformation      5 6 10
Householder, A. S.      4

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