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| Berman A., Plemmons R.J. — Nonnegative matrices in the mathematical sciences |
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| Предметный указатель |
 -inverse 117
-monotone matrix 119
 -positive matrix 22
 -matrix 160
 -matrix 160 164
Aggregation/disaggregation methods 312
Algebraic eigenspace 42
Almost monotone matrix 120
Assumption of fixed proportion of factor inputs 249
Asymptotic rate of convergence 172
Birkhoff theorem on doubly stochastic matrices 50
Birkhoff theorem on K-nonnegative matrices 6
Bounds for eigenvalues not equal to the spectral radius 51 59
Chain of classes 42
Chain of classes, length of 42
Circuit 34
class 39
Class, basic 39
Class, final 39
Class, height of 39
Class, nonbasic 39
Cogredience 27
Collatz — Wielandt function 60
Communication, between classes 39
Communication, between indices 39
Comparision matrix 142
Complementary pivot algorithm 295—296
Completely decomposable matrix 75
Completely monotonic functions 142—146
Completely positive matrix 20 304—306
Completely reducible matrix 53
Cone 2 21
Cone independence 71
Cone, complementary 276
Cone, convex 2
Cone, dual 1
Cone, ice cream 2
Cone, indecomposable 20
Cone, pointed 2
Cone, polyhedral 2
Cone, proper 3 181
Cone, reproducing 2
Cone, simplicial 3
Cone, solid 2
Convergent matrix 9
Convex body 5
Copositive matrix 20 295
Copositive matrix with respect to a cone 23
Copositive matrix, strictly 295
Copositive plus matrix 295
cp-rank 304
Cross-positive matrix 21
Cross-positive matrix, strongly 21
Cyclic matrix 34 309—310
D-monotone 119
DAD theorems 58—59
Degree of a nonnegative matrix 42
Degree of an eigenvalue 6
Diagonal of a matrix 49
Diagonally dominant matrix 20
Diagonally dominant matrix, generalized column 186
Diagonally dominant matrix, irreducibly 186
Diagonally dominant matrix, lower semistrictly 137
Diagonally dominant matrix, strictly 137 186
Directed graph 29
Directed graph, strongly connected 30
Distribution vector, initial 213
Distribution vector, initial probability 212
Distribution vector, probability 212
Distribution vector, stationary 213
Doubly nonnegative matrix 304—306
Doubly stochastic, matrix 48 66 240
Doubly stochastic, pattern 55
Doubly substochastic matrix 104
Drazin inverse 118 147 198 227
Eigenset 191
Eigenvector, generalized 42
Eigenvector, principal 7
Equimodular matrices 142
Ergodic state 219
Essentially nonpositive matrices 252 see of "Z-matrices"
Extreme points of, matrix polytopes 100—105
Extreme points of, polytope of doubly stochastic matrices see "Birkhoff theorem"
Face 3
Factorizable matrix 75
Finite homogeneous Maikov chain see "Markov chain"
Fully indecomposable matrix 56
Fundamental matrix associated with a Markov chain 234
Gauss — Seidel method 171
Gauss — Seidel method, block 174
Gauss — Seidel method, point 174
Generalized left inverse 118
Generalized left inverse positive matrix 121
Generalized right inverse 118
Generalized ultrametric matrices 307
Green's relations 64
Green's relations on the semigroup of nonnegative matrices 72—73
Group inverse 118 226—240
Group monotone matrix 119
H-matrix 185
Hadamard product 159
Hadamard — Fischer inequality 57—160
Hawkins — Simon condition 161 253
Heat equation 167
Hitchcock's transportation problem 110
Idempotent nonnegative matrices 65—67
Incidence matrix 75
Indecomposable matrix see "Irreducible matrix"
Index of a square matrix 61 118 198
Index of cyclicity 34
Index of primitivity 46
Index of primitivity, bounds for 46—48
Input matrix 251—257 260—266
Input-output table 247
Inverse eigenvalue problem 61 87
Inverse eigenvalue problem for primitive matrices 302—303
Inverse eigenvalue problem for symmetric nonnegative matrices 88—92 301 303
Inverse eigenvalue problem for symmetric positive matrices 92—93
Inverse eigenvalue problem with prescribed diagonal elements 93—98
Inverse M-matrix problem 163
Inverse nonnegative matrix 306—307
Inverse positive matrix 137
Irreducible M-matrix 141 156 160 267
Irreducible matrix 27 29—38
Irreducible monotone matrix 116
Jacobi method 170
Jacobi method, block 174
Jacobi method, point 174
K-inverse-positive matrix 112
K-irreducible matrix 11—13
K-monotone matrix 112
K-nonnegative matrix 5
K-nonsingular M-matrix 112
K-positive matrix 5 13—14
K-primitive matrix 16—18
K-reducible matrix 11
K-semipositive matrix 9 114
K-singular M-matrix 114
K-stochastic matrix 23
Krein — Rutman theorem 6
L-matrix 296
Least element 279
Left inverse 67
Leontief input-output model 243
| Leontief input-output model, closed 245 258—265
Leontief input-output model, feasibility of 260
Leontief input-output model, open 245 251—258
Leontief input-output model, open, feasibility of 252
Leontief input-output model, open, profitability of 252
Leontief matrix 282 296
Leontief matrix, pre 282
Leontief matrix, totally 282
Lexico-positive vector 274
Limiting matrix of a Markov chain 227 234
Linear complementarity problem 271
Linear complementarity problem with upper bounds 281
Linear complementarity problem, over cones 298
Linear complementarity problem, parametric 281
Linear equations 165
Linear least-squares problem 178
LP solvability 285—290
LU decomposition of an M-matrix 157
LU nonnegative factorization 86
Lyapunov theorem 162
M-matrix 133
M-matrix with property c 152—156 200—202 226—228 261 263
M-matrix, nonsingular 133—142 158—164 166 181—190 245 253—255 267—270 272 290—292 294
M-matrix, singular 147—164 207 213 242 310—311
Markov chain 48 212—243
Markov chain, absorbing 221
Markov chain, ergodic 220
Markov chain, period of 221
Markov chain, periodic 220
Markov chain, regular 220
Markov chain, solutions of using iterative methods 307—312
Matrices of Leontief type 252
Maximal subgroups of nonnegative matrices 71
Monomial matrix 67
Monotone matrix 137
Monotone matrix in a set 119
Moore — Penrose generalized inverse 83 86 119—127 207 240
Nearly completely decomposable matrix 312
Nonlinear complementarity problem 298
Nonnegative matrix 26—62
Nonnegative matrix on a set 119
Nonnegative matrix, doubly 304—306
Nonnegative matrix, inverse 306—307
Nonnegative orthant 1
Nonnegative rank 304
Nonnegative rank factorization 67 84
Nonnegative stability 150
Normal form of a cyclic matrix 32
Normal form of a reducible matrix 43
Open sector 244
Optimum SOR relaxation parameter 174
Orthostochastic matrix 85
Oscillatory matrix 57
Ostrowski — Reich theorem 195
Output equilibrium vector 261
P-matrix 134 272—276
Partly decomposable matrix 75
Permanent 51
Perron root 299—301
Perron — Frobenius theorem 27 32
Perron — Frobenius theorem in a game of numbers 312—314
Perron — Frobenius theorem in nonnegative linear systems 314—315
Poisson's equation 196
Polyhedron 49
Polytope 49
Positive definite matrix 20
Positive eigenvector of an irreducible matrix 37 60
Positive matrix 26
Positive semidefinite matrix 20
Positive stable matrix 113 135
Price vector 252
Prime matrix 75—82
Primitive matrix 28 45—54 223—226
Principal minors 34—35 149—150
Principal pivot transform 273
Principal pivoting method 271—272
Proper splitting 207
Q-matrix 276—279
Quadratic form sign equivalent hermitian matrices 192
r-monotone matrix 67
Random walk 217 236—241
Rank factorization 67
Rectangular monotone matrix 119
Reduced triangular block form 262 see
Reducible matrix 27 38—45
Regular D-class 64 70
Regular element 64
Regular matrix 277
Regular progression hypothesis 291
Regular splitting 130
Regular splitting, convergent 138
Regular splitting, convergent weak 138
Regular splitting, theorem 183
Regular splitting, weak 151
Restricted basis simplex method 291
Right inverse 67
Safety factor 291
Schur complement 159 293
Schur's determinantal formula 293
Semi-inverse 64—117
Semiconvergent matrix 152 197 228—229 235—237
Semigroup 64
Semigroup of doubly stochastic matrices 82—83
Semigroup of nonnegative matrices 67—82
Semimonotone matrix 119
Semimonotone matrix, strictly 277
Semipositive matrix 136
Set monotonicity 127—128
Simple odd cactus 101
Simple path 34
Simple path, length of 34
Singular linear system 196—203
SOR method 169 172—181
SOR method, block 174
SOR method, p-cyclic 307—312
SOR method, point 174
SOR method, symmetric 204
Spectral radius 6 31 52 53 56 159
Spectral radius, bounds for 37 38 60
Star-shaped set 291
Stationary probability distribution vector 212—241
Stein — Rosenberg theorem 187 201
Stieltjes matrix 109 141 201 221—224 265
Stochastic matrix 48—52 54 212 314
Stoper — Samuelson, condition 270
Stoper — Samuelson, theorem 270
Strictly ultrametric matrices 307
Subpermutation matrix 104
Successive overrelaxation method see "SOR method"
T-monotone matrix 120
Taussky's unification problem 164
Term rank 110
Term rank, symmetric 110
Totally nonnegative matrix 57 163
Totally positive matrix 57
Transient state 219
Transition matrix 212
Transition matrix, standard form 224
Triangular block form 39 54 see
Value added vector 252
Van der Waerden conjecture 51 62
von Neumann model 269
Weak monotone 120
Z-matrix 279
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