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| Авторизация |
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| Horn R.A. — Matrix Analysis |
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| Предметный указатель |
a priori bounds 337
Absolute convergence 279 300
Absolute value of a complex number 532
Absolute vector norm 285 310 365 438
Additive property, of inner product 260
Adjacency matrix, of a graph 168 523
Adjoint, classical 20
Adjoint, Hermitian 6
Adjugate 20
Algebraic multiplicity 58 60 138 141 497 499
Algebraically closed field 41 537
Algebraically simple eigenvalue 371 500 508
Alternating sum 8
Angle between vectors 15
Annihilate 142
Antilinear transformation 250
Approximation problems 332 427
ARC 357
arg 532
Argument of a complex number 532
Arithmetic-geometric mean inequality 535
Augmented matrix 11 12
Back substitution 159
Backward identity 28 207
Ball of radius r 281 541
Basis 3
Basis, change of 30
Basis, orthonormal 16
Basis, representation 31
Bilinear form 169 175
Biorthogonality 59
Birkhoff’s theorem 197 527
Block, diagonal matrices 24
Block, triangular matrices 25
Bochner’s theorem 394
Bordered matrix 185
Bound norm 294
Boundary 282
Bounded set 282 541
Brauer, condition for invertibility 381
Brauer, region 380
Brauer, theorem 380
Brualdi, condition for invertibility 389
Brualdi, region 385
Brualdi, theorem 385 387
Cancellation theorem 78 141
Canonical forms, consimilarity 251
Canonical forms, integer matrices 158
Canonical forms, irreducible normal form 506
Canonical forms, Jordan 121
Canonical forms, polar 156 412ff
Canonical forms, rational 154
Canonical forms, rational canonical 156
Canonical forms, rational matrices 158
Canonical forms, real Jordan 152
Canonical forms, real orthogonal matrices 108
Canonical forms, real skew-symmetric matrices 107
Canonical forms, real symmetric matrices 107
Canonical forms, singular value decomposition 157 414ff
Canonical forms, symmetric Jordan 209
Canonical forms, triangular factorization 157
Carmichael and Mason’s bound on zeroes 317 318 364
Cassini, ovals of 380
Cauchy bound on zeroes 316 318 364
Cauchy sequence 274
Cauchy — Binet formula 22
Cauchy — Schwarz inequality 15 261 277 535 536
Cayley — Hamilton theorem 86
Cesaro summation 524
Change of basis matrix 32 33
Characteristic equation 87
Characteristic polynomial 38 86 87 540
Characteristic polynomial, Cayley — Hamilton theorem 86
Characteristic polynomial, definition 38
Characteristic polynomial, positive definite matrix 403
Cholesky factorization 114 407
Circulant matrices 26
Classical adjoint 20
Closed set 282 541
Closure 282
Cofactor 17
Column rank 12
Combinatorially symmetric matrix 523
Commutative ring 95
Commutator 98
Commuting family 51 81 99 139
Commuting matrices 135
Compact set 282 541
Companion matrix 147 149 316 514
Compatible vector norm 294 324 327
Completeness property of a vector space 274
Complex conjugate 531
complex numbers 531 532
Compound matrix 19
Concave function 534
Concave logarithm of determinant 466
Concave trace of inverse 468
Condiagonalization 244 248
Condition number 336 340 365 366 374 442
Coneigenvalue characterization 246
Coneigenvalue definition 245
Coneigenvector 245
conformal 17
Congruence,  220
Congruence, *congruence 220 399 464ff 470
Congruence, simultaneous *congruence, canonical pairs 236
Conjugate linear 169
Conjunctive 220
Consimilarity 234 244
Consimilarity, characterizations 251
Consimilarity, to a real matrix 255
Consistent linear equations 12
Consistent vector norm 324
Constrained extrema 34
Continuous dependence of eigenvalues 540
Continuous dependence of zeroes 539
Continuous function, maximum of a 541
Contriangularization 244
Convergence of a sequence 269
Convergent matrix 137 298
Convex 284
Convex, combination 535
Convex, cone 463
Convex, function 392 533 534—536
Convex, hull 533
Convex, metric hull 289
Convex, sets 533—536
Coordinate representation 30
Correlation matrix 400
Courant — Fischer theorem 179 420 424 472
Covariance matrix 219 239 392 424
Cramer’s Rule 21
CYCLE 357
Cyclic of index k 514
Defect from normality 316
Defective 58
Deflation 63 83
Deleted absolute row sums 344
Dependent 3
Determinant 7 11 398
Determinantal inequalities 453 467 476—486
Determinantal inequalities, Fischer 478
Determinantal inequalities, Gersgorin 351
Determinantal inequalities, Hadamard 477
Determinantal inequalities, Hadamard — Fischer 485
Determinantal inequalities, Minkowski 482
Determinantal inequalities, Oppenheim 480
Determinantal inequalities, Ostrowski — Taussky 481
Determinantal inequalities, Szasz 479
Diagonal entries, equal 77
| Diagonal matrix 23
Diagonalizable 139 145
Diagonalizable, almost 89
Diagonalizable, by orthogonal similarity 211
Diagonalizable, definition 46
Diagonalizable, orthogonally 101
Diagonalizable, simultaneously 49
Diagonalizable, unitary 101
Diagonalization by congruence 228
Diagonalization by consimilarity 234 244 248
Diagonalization by similarity 46 145
Diagonalization by unitary congruence 204
Diagonalization by unitary consimilarity 244 245
Diagonalization by unitary similarity 101
Diagonalization, simultaneous 52
Diagonally dominant 349
Diagonally dominant, strictly 302 349
Difference scheme 394
Differential equations 132 394
Differential equations, elliptic 239 459
Differential equations, hyperbolic 239
Differential equations, partial 168 216 218
DIMENSION 4
Direct sum 24
Directed graph of a matrix 357 517 522
Directed path 357
Dominant eigenvalue 506
Doubly stochastic matrix 197 527—529
Dual norm 275 410
Dual norm of Euclidean norm 277
Dual pair 278
Duality theorem 287
Edges 168
Eigenspace 57
Eigenvalue, algebraic multiplicity 58 60 138 141 497 499
Eigenvalue, algebraically simple 371
Eigenvalue, continuous dependence on matrix entries 540
Eigenvalue, definition 35
Eigenvalue, deflation to calculate 63
Eigenvalue, distinct 48
Eigenvalue, dominant 506
Eigenvalue, generalized 213
Eigenvalue, geometric multiplicity 58 60 138 141 497 498
Eigenvalue, ill-conditioned 367
Eigenvalue, inclusion region 501
Eigenvalue, inclusion theorem 177
Eigenvalue, index of 131 139 148
Eigenvalue, location 343
Eigenvalue, moments 43
Eigenvalue, nonnegative matrix 489
Eigenvalue, of a sum 181 184
Eigenvalue, perfectly conditioned 367
Eigenvalue, perturbation 198 343 364
Eigenvalue, positive definite matrix 402
Eigenvalue, positive matrix 496
Eigenvalue, power method to calculate 62
Eigenvalue, principal submatrices 189
Eigenvalue, variational characterization 176—180
Eigenvalue, well-conditioned 367
Eigenvector 57
Eigenvector, definition 35
Eigenvector, left 59 371
Eigenvector, positive 493 494 495 513
Eigenvector, right 59
Elementary divisors 155
Elementary symmetric functions 41
Elliptic differential operator 239
Equilibrated 283
Equivalence relation, congruence 221
Equivalence relation, consimilarity 251
Equivalence relation, definition 45
Equivalence relation, vector seminorm 262
Equivalent matrices 164
Equivalent vector norms 273 279
Equivalent, orthogonally 73
Equivalent, real orthogonally 73
Equivalent, unitarily 72
Error analysis 335
Essentially nonnegative matrix 506
Essentially triangular matrix 26
Euler’s theorem 111
Exponential of a matrix 300
Extreme points, closed convex set 533
Extreme points, doubly stochastic matrices 528
Extreme ray, definition 463
Extreme ray, positive semidefinite matrices 464
Factor analysis 431
Factorizations 156
Factorizations, Cholesky 114 407
Factorizations, complex skew-symmetric matrix 217
Factorizations, complex symmetric matrix 204
Factorizations, LU 158—165
Factorizations, polar 156 411 412ff
Factorizations, product of two Hermitian matrices 172
Factorizations, QR 112 164 406
Factorizations, singular value decomposition 411
Factorizations, Takagi 250 423 466
Factorizations, triangular 157
Family, commuting 51 81 99 139
Family, commuting real normal 108 112
Family, complex symmetric 243
Family, diagonalizable symmetric 217
Family, Hermitian 172
Family, normal 103
Family, simultaneous condiagonalization 252
Family, simultaneous diagonalization by *congruence 239
Family, simultaneous diagonalization by unitary 243
Family, simultaneous singular value decomposition 426
Family, simultaneous triangularization 84
Fan, k norms 445
Fan, theorem on eigenvalue location 501
Fejer, trace theorem, on positive semidefinite matrices 459
Fejer, uniqueness theorem, for elliptic partial differential equations 460
Field 1
Field of values 321 332
Fischer’s inequality 478
Forms, bilinear 169 175
Forms, hermitian 174
Forms, quadratic 168 174 214 466
Forms, sesquilinear 169
Forward substitution 159
Fundamental theorem of algebra 537
General linear group 14
Generalized coordinates 227
Generalized eigenvalue 213
Generalized inverse 421
Generalized matrix functions 8
Generalized matrix norm 290 320 see
Geometric multiplicity 58 60 138 141 497 498
Gersgorin circles 346
Gersgorin disc theorem 344
Gersgorin discs 345 353
Gersgorin region 345
Givens’s method 77
Gram matrix 407
Gram — Schmidt process 15 148
Gram — Schmidt process, modified 116
Gram — Schmidt process, symmetric analogue 211
Graph 168
Greub and Rheinboldt inequality 452
Group, finite abelian 510
Group, general linear 14
Group, isometry 266 267
Group, orthogonal 69 71
Group, unitary 69
Grunsky inequalities 202
Hadamard — Fischer inequalities 485
Hadamard, exponential of a matrix 461
Hadamard, inequality 199 200 477 483
Hadamard, powers of a matrix 462
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