Berman A. — Nonnegative Matrices in the Mathematical Sciences :: Электронная библиотека попечительского совета мехмата МГУ

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Berman A. — Nonnegative Matrices in the Mathematical Sciences

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Название: Nonnegative Matrices in the Mathematical Sciences

Автор: Berman A.

Аннотация:

Here is a valuable text and research tool for scientists and engineers who use or work with theory and computation associated with practical problems relating to Markov chains and queuing networks, economic analysis, or mathematical programming. Originally published in 1979, this new edition adds material that updates the subject relative to developments from 1979 to 1993. Theory and applications of nonnegative matrices are blended here, and extensive references are included in each area. You will be led from the theory of positive operators via the Perron-Frobenius theory of nonnegative matrices and the theory of inverse positivity, to the widely used topic of M-matrices.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 340

Добавлена в каталог: 11.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$\lambda$ -inverse      117
$\lambda$ -monotone matrix      119
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 HT-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—2
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 also Matrices of
Extreme points of matrix polytopes      100—105
Extreme points of matrix polytopes, polytope of doubly stochastic matrices      (see Birkhoff theorem)
Face      3
Factorizable matrix      75
Finite homogeneous Markov 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, 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-matiix 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 15
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, 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
M-positive matrix      22
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      67 137
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
O-matrix      160
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 also Normal form)
Reducible matrix      27 38—45
Regular element      64
Regular matrix      277
Regular O-class      64 70
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-matrix      160 164
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
Value added vector      252
Van der Waerden conjecture      51 62
von Neumann model      269
Weak monotone      120
Z-matrix      279

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