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| Stewart G.W., Sun J. — Matrix perturbation theory |
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| Предметный указатель |
 in terms of the inverse matrix 36
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  -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,  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  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 theorem 251 255 258
Hermitian matrix, skew Hermitian matrix 5
Hermitian matrix, spectral decomposition 19 226
Hermitian matrix, sums of 27
Hermitian matrix, tan 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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