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| Авторизация |
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| Horn R.A. — Matrix Analysis |
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| Предметный указатель |
Modulus 532
Moment sequence, Hausdorff 393
Moment sequence, Toeplitz 393
Moments of eigenvalues 43 44
Moments, algebraic 393
Moments, trigonometric 393 455 456
Monic polynomial 142
Monotone vector norm 285 310 450
Monotonicity theorem 182
Montel’s bound on zeroes 317 318 364
Moore — Penrose generalized inverse 421
Multilinear 11
Multiplicity, algebraic 58 60 138 141 497 499
Multiplicity, geometric 58 60 138 141 497 498
Negative, definite matrix 397
Negative, semidefinite matrix 397
Nehari’s theorem 202
Newton’s identities 44
Nilpotent 37 139
nodes 168
Nondefective 58 103
Nonderogatory 58 135 147
NonNegative 259 260 290
Nonnegative matrix 359
Nonnegative matrix, applications 487—490
Nonnegative matrix, definition 490
Nonnegative matrix, doubly stochastic 527—529
Nonnegative matrix, eigenvalues 489 503—505 507—515
Nonnegative matrix, eigenvectors 489 503—505 507—515
Nonnegative matrix, general limit theorem 524
Nonnegative matrix, irreducible 507—515
Nonnegative matrix, limit of powers 489
Nonnegative matrix, limit theorems 500 516 524 525
Nonnegative matrix, Perron root 505 508
Nonnegative matrix, Perron vector 505 508
Nonnegative matrix, Perron — Frobenius theorems 508—511
Nonnegative matrix, primitive matrices 515—524
Nonnegative matrix, spectral radius 489 491—495 503—505 507—515
Nonnegative matrix, stochastic 526—529
Nonsingularity, characterizations of 14
Nontrivial cycle 383
Norm, and inversion 301
Norm, characterization of derived 263
Norm, compatible 294 324
Norm, consistent 324
Norm, dual 275 410
Norm, dual pair 278
Norm, minimally spectrally dominant 330
Norm, pre-norm 272
Norm, spectrally dominant 324
Norm, subordinate 324
Normal matrix, characterizations 101 108—112
Normal matrix, definition 100
Normal matrix, perfectly conditioned eigenvalues 367
Normal matrix, real 104ff
Normalized vector 15
Null space 5 262
Numerical radius 321 331 332 333 334
Numerical range 321
Open set 282 541
Operator norm 294
Oppenheim’s inequality 480
Orthogonal complement 16
Orthogonal group 69 71
Orthogonal matrix 157
Orthogonal matrix, complex 71 72
Orthogonal matrix, real 66 72
Orthogonal matrix, skew 72
Orthogonal vectors 15
Orthogonality 15
Orthogonally diagonalizable 101
Orthogonally equivalent 73
Orthonormal 15
Orthonormal basis 16
Orthostochastic matrices 197
Ostrowski condition for invertibility 381
Ostrowski region 378 379
Ostrowski theorem 224 378
Ostrowski — Taussky inequality 468 481
Ovals of Cassini 380
Parallelogram identity 263
Partitioned matrix, definition 17
Partitioned matrix, inverse 18 472
Partitioned matrix, Schur complement 472
Pearcy’s theorem 76
Perfectly conditioned 336
Permanent 8
Permutation 368
Permutation, invariant 438
Permutation, matrix 25 360
Perron Perron — Frobenius theorems 508—511
Perron root 497 505 508
Perron theorem 500
Perron vector 497 505 508
Perturbation, eigenvalues 198 343 364
Perturbation, linear equations 335
Perturbation, theorems 364
Plane rotations 74
Poincare separation theorem 190 441
Polar coordinates in complex plane 532
Polar decomposition see "Polar form"
Polar form 156 411 412ff
Polar form, examples and applications 427ff
Polarization identity 263
Polynomial, bounds for zeroes of 316—319
Polynomial, characteristic 38
Polynomial, continuous dependence of zeroes on coefficients 539
Polynomial, for inverse 88
Polynomial, in a matrix 36 135 142
Polynomial, monic 142
Polynomial, similarity invariants 154
Polynomial, zeroes of a 537
Polynomial, zeroes of a real 538
Poorly conditioned 336
Positive 259 260 290
Positive definite matrix 250
Positive definite matrix, applications 391—396 459
Positive definite matrix, characteristic polynomial 403
Positive definite matrix, characterizations 402
Positive definite matrix, concavity of logarithm of the determinant 466
Positive definite matrix, concavity of trace of the inverse 468
Positive definite matrix, definition 396
Positive definite matrix, determinant criterion 404
Positive definite matrix, determinantal inequalities 476—486
Positive definite matrix, eigenvalues 402
Positive definite matrix, kth root 405
Positive definite matrix, ordering 469ff
Positive definite matrix, square root 405
Positive matrix 359
Positive matrix, definition 490
Positive matrix, eigenvalues 495—503
Positive matrix, eigenvectors 495—503
Positive matrix, left Perron vector 497
Positive matrix, Perron root 497
Positive matrix, Perron vector 497
Positive matrix, Perron’s theorem 500
Positive matrix, spectral radius 495—503
Positive semidefinite matrix 182
Positive semidefinite matrix, definition 396
Positive semidefinite matrix, kth root 405
Positive semidefinite matrix, ordering 469ff
Positive semidefinite matrix, rank k 457
Positive semidefinite ordering 469
Positive, cone 398
Positive, definite function 400 401 463
Positive, definite kernel 402 462
Power inequality 333 334
Power method 62 523
Pre-norm 272 322
Preorder 384
| Primitive matrix 515—524
Primitive matrix, characterizations 516 517
Primitive matrix, definition 516
Primitive matrix, directed graph 517
Primitive matrix, eigenvalues 516
Primitive matrix, Holloday — Varga theorem 520
Primitive matrix, index of primitivity 519
Primitive matrix, Wielandt’s theorem 520
Principal minor 17 40
Principal submatrix 17
Principal submatrix, eigenvalues 189
Procrustean transformation 431
Property, L 97 100
Property, P 100
Property, SC 355 358 359 362
Pure imaginary complex number 532
QR, algorithm 114
QR, factorization 112 164 406
Quasi-linearization 191 453 455 486
RANGE 5
Rank 12
Rank one, approximation 427
Rank one, limit 499
Rank one, matrix 61
Rank, equalities 13
Rank, inequalities 13 175 352 458
Rational canonical form 156
Rational form 154
Rayleigh — Ritz ratio 176
Rayleigh — Ritz theorem 176 422
Real axis 532
Real Jordan canonical form 152
Real part of a complex number 531
Rectangular coordinates in complex plane 532
Reducible matrix 360
Residual vector 338 373 374
Reverse order law 6
Riemann sum 462
Right, eigenvector 59
Right, half-plane 532
Robertson’s inequality 468
Romanovsky’s theorem 517
Rotation problem 431 435
Round-off errors 335
Row rank 12
Row-reduced echelon form (RREF) 10
Scalar matrix 23
Scalar product 14
Schatten p norms 441
Schur, complement 21 472
Schur, majorization theorem 193
Schur, norm 291
Schur, product see "Hadamard product"
Schur, product theorem 455 458
Schur, unitary triangularization theorem 79 83
Selection principle, for unitary matrices 69 117 416
Self-adjoint, matrix norm 309
Self-adjoint, vector norm on matrices 450
Seminorm 259
Sensitivity of eigenvalues 372
Sensitivity of eigenvectors 373
Sensitivity of solutions of linear equations 339
Separating hyperplane theorem 534
Separation theorem 190
Sesquilinear forms 169
Shift operator 38
Signature of a matrix 221
Signum (sgn) 8
Similarity 44
Similarity, inverse to adjoint 70
Similarity, matrix and its transpose 134
Similarity, to a real matrix 172
Similarity, to a real matrix,  253
Similarity, to adjoint 172
Simultaneous *congruence, canonical pairs 236
Simultaneous condiagonalization 252
Simultaneous diagonalization 49 52
Simultaneous diagonalization by *congruence 240 464ff
Simultaneous diagonalization by congruence 228 241 250
Simultaneous diagonalization by unitary congruence 228 235
Simultaneous diagonalization, characterization of 228
Simultaneous singular value decomposition 426
Simultaneous triangularization 81 84 94
singular 14
Singular value decomposition 157 205 325 411 414ff 421
Singular value decomposition, examples and applications 427ff
Singular value decomposition, simultaneous 426
Singular values 205 415
Singular values, largest 421
Singular values, of a product 423
Singular values, of a sum 423
Singular values, perfectly conditioned 418
Singular values, variational characterization 420
Skew orthogonal 72
Skew-Hermitian matrix 100 169
Skew-Hermitian part 109 170 399
Skew-symmetric matrix 397
Solution-equivalent systems 11
span 2 3
Specht’s theorem 76
Spectral characteristic 330
Spectral norm 295 308 309 421 441 445 450
Spectral radius 35 198 296 313 348 489
Spectral radius as a limit of matrix norms 297
Spectral radius as a limit using norms or pre-norms 299 322
Spectral Theorem, Hermitian matrices 104 171
Spectral theorem, normal matrices 101 425
Spectrally dominant 329
Spectrum 35
Spectrum of a sum 92
Spectrum of a sum by majorization 194
Square root of a matrix 54 405
Square root of a matrix, continuity of 411
Standard basis 4
Standard inner product 14
Stochastic matrix 526—529
Strictly diagonally dominant 302 349
Strongly connected directed graph 358 362 383
SubMatrix 4
Submatrix, eigenvalues 189
Submatrix, principal 17 397
Submultiplicative 290
Subordinate vector norm 324
subspace 2
Subspace, invariant 51
Sylvester’s Identity 22
Sylvester’s law of inertia 223 238
Sylvester’s law of inertia, analogue for symmetric matrices 225
Sylvester’s law of inertia, homotopy proof 242
Symmetric gauge function 438 445
Symmetric Jordan canonical form 209
Symmetric matrices,  225
Symmetric matrix 167 397
Symmetric matrix, complex 201
Symmetric matrix, diagonalizable 211
Symmetric matrix, every matrix similar to a 209
Symmetric matrix, product of two 210
Symmetric matrix, real 169 218
Szasz’s inequality 479
Takagi’s factorization 204 423 466
Takagi’s factorization as consimilarity analogue of spectral theorem 250
Takagi’s factorization, Hua’s proof 217
Takagi’s factorization, Siegel’s proof 216
Taussky’s theorem 363
Toeplitz matrix 27 136 137 394 409 456 462
Toeplitz moment sequence 393
Topological notions 282 288
Trace 40 175 398
Trace, norm 441 445
Trace, zero 77
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