Berman A., Shaked-Monderer N. — Completely Positive Matrices :: Электронная библиотека попечительского совета мехмата МГУ

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Berman A., Shaked-Monderer N. — Completely Positive Matrices

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Название: Completely Positive Matrices

Авторы: Berman A., Shaked-Monderer N.

Аннотация:

A real matrix is positive semidefinite if it can be decomposed as A=BB¢. In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A=BB¢ is known as the cp-rank of A.
This invaluable book focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp-rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 206

Добавлена в каталог: 23.04.2010

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

Предметный указатель

$\widehat{CP}$ graph      107
2-norm      2
Adjacency matrix      37
Angle between vectors      2
ARC      38
Block      33
Block design, balanced incomplete      69
Block design, complete      69
Block design, incidence matrix of      69
Book graph      101
Caratheodory's theorem      46
Cauchy — Binet formula      4
Cayley's formula      40
Characteristic polynomial      6
Chord      32
Chromatic number      36
CLIQUE      32
Clique number      36
Clique, maximal      32
Completely positive graph      83
Completely positive matrix      61
Completely positive matrix, cp-rank of      139
Completely positive matrix, critical      67
Completely positive matrix, factorization index of      146
Completely positive matrix, i-minimizable      67
Completely positive matrix, rank 1 representation of      64
Completely positive quadratic form      68
Completion      56
Completion, CP      133
Completion, PSD      57
Component      33
Cone      41
Cone, circular      42
Cone, convex      42
Cone, dual      46
Cone, full      48
Cone, pointed      48
Cone, polyhedral      49
Cone, relative boundary of      44
Cone, relative interior of      44
Cone, reproducing      48
Cone, solid      48
Convex set      41
Copositive matrix      51
CP completion      133
cp-rank      139
Crown      97
Cut vertex      33
CYCLE      32
Cycle, odd      34
Diagonally dominant matrix      7
Diagonally dominant matrix, irreducibly      7
Diagonally dominant matrix, strictly      7
Digraph      38
Digraph, strongly connected      38
Direct sum of matrices      3
Directed graph      38
Directed graph of a matrix      38
Distance between vertices      32
DJL Conjecture      157
Dog-ear      122
Doubly nonnegative matrix      20
Doubly nonnegative matrix, diminishable      114
EDGE      30
Edge, adjacent      31
Edge, end of      31
Edge, incident      31
Edge, vertex of      31
Edge-clique cover number      144
Eigenspace      6
Eigenvalue      6
Eigenvalue, simple      6
Eigenvector      6
Euclidean vector space      1
Excellent graph      122
Factorization index      146
G-partial matrix      56
G-partial matrix, CP      134
G-partial matrix, DNN      136
G-partial matrix, PD      60
G-partial matrix, PSD      57
Gersgorin's theorem      6
Graph      30
Graph, $CP^{1}$       175
Graph, $CP^{2}$       175
Graph, $\widehat{CP}$       107
Graph, bipartite      34
Graph, block of      33
Graph, block-clique      35
Graph, book      101
Graph, chordal      34
Graph, complement of      31
Graph, complete      31
Graph, complete bipartite      34
Graph, completely positive      83
Graph, connected      32
Graph, disconnected      32
Graph, excellent      122
Graph, isomorphic      31
Graph, k-colorable      36
Graph, k-connected      33
Graph, line graph of      37
Graph, line perfect      37
Graph, NCC      117
Graph, NLOC      97
Graph, PD completable      60
Graph, perfect      36
Graph, PSD completable      58
Graph, root graph of      37
Graph, simple      30
Graph, triangle free      34
Hadamard function of a matrix      130
Hadamard power      14
Hadamard product      13
Halfspace, closed      44
Halfspace, open      44
Hyperplane      44
Hyperplane, separating properly      44
Index of cyclicity      19
Inertia      7
Inner cycle of $S_{2n}$       181
Krein — Milman theorem      45
Kronecker product      14
Laplacian matrix      37
LU-completely positive matrix      123
M-matrix      21
M-matrix, inverse      127
Matrix realization, CP      83
Matrix realization, DNN      83
Matrix realization, symmetric      39
Matrix realization, symmetric nonnegative      39
Matrix, adjacency      37
Matrix, Cauchy      5
Matrix, Cauchy, symmetric      5
Matrix, comparison      3
Matrix, completely positive      61

Matrix, copositive      51
Matrix, diagonally dominant      7
Matrix, diminishable doubly nonnegative      114
Matrix, doubly nonnegative      20
Matrix, doubly stochastic      127
Matrix, excellent      122
Matrix, G-partial      56
Matrix, G-partial CP      134
Matrix, G-partial DNN      136
Matrix, G-partial PD      60
Matrix, G-partial PSD      57
Matrix, Gram      11
Matrix, Hadamard function of      130
Matrix, Hankel      72
Matrix, Hilbert      5
Matrix, infinitely divisible      131
Matrix, irreducible      3
Matrix, irreducibly diagonally dominant      7
Matrix, Laplacian      37
Matrix, LU-completely positive      123
Matrix, nonnegative      18
Matrix, partial      56
Matrix, partial CP      133
Matrix, partial DNN      136
Matrix, partial PSD      57
Matrix, permutation      2
Matrix, positive      18
Matrix, positive definite      15
Matrix, positive diagonal      2
Matrix, positive semidefinite      9
Matrix, primitive      23
Matrix, reducible      3
Matrix, Stieltjes      24
Matrix, stochastic      127
Matrix, strictly diagonally dominant      7
Matrix, totally nonnegative      20
Matrix, totally positive      20
Matrix, triangular totally nonnegative      21
Matrix, triangular totally positive      21
Matrix, UL-completely positive      123
Matrix-tree theorem      38
Menger's Theorem      40
Minor, bordered      5
Minor, leading principal      3
Minor, left almost principal      124
Minor, principal      3
Minor, right almost principal      124
Moore — Penrose generalized inverse      4
n-cycle      32
NCC graph      117
Nested principal minors      16
NLOC graph      97
Non crossing cycle      122
Nonnegative matrix      18
Nonnegative orthant      1
Nonnegative rank      146
Nonnegative vector      1
Normalization      137
Partial CP matrix, meets the cycle condition      137
Partial CP matrix, normalization of      137
Partial DNN matrix      136
Partial DNN matrix, meets the cycle condition      137
Partial matrix, CP      133
Partial matrix, PSD      57
Partial matrix, symmetric      56
Path      32
Path, directed      38
Path, internal vertex of      32
PD completable graph      60
Perfect elimination ordering      35
Perron root      19
Perron vector      19
Perron — Frobenius theorem      19
PLSS property      98
Positive definite matrix      15
Positive diagonal congruence      3
Positive matrix      18
Positive semidefinite matrix      9
Positive semidefinite matrix, i-minimized      67
Positive semidefinite matrix, square root of      12
Positive vector      1
Property PLSS      98
PSD completable graph      58
PSD completion      57
Rank 1 representation      64
Rank 1 representation of support k      82
Rank 1 representation, minimal      139
Ray      45
Ray, extreme      45
Schur complement      24
Schur complement, generalized      27
Schur product      18
Separation, proper      44
Specification graph      56
Spectral radius      6
Spectrum      6
Square root of a matrix      12
Standard basis      2
Standard inner product      2
Strong perfect graph conjecture      41
Subgraph      31
Subgraph, induced      31
Subgraph, spanning      31
Subgraphs, disjoint      32
Subgraphs, intersection of      32
Subgraphs, union of      32
Submatrix, leading principal      3
Submatrix, principal      3
Support      1
Sylvester's Determinantal Identity      4
Sylvester's law of inertia      8
Totally nonnegative matrix      20
Totally positive matrix      20
TREE      33
triangle      32
Triangular totally nonnegative matrix      21
Triangular totally positive matrix      21
Turan's theorem      40
UL-completely positive matrix      123
Vector, extreme      45
Vector, nonnegative      1
Vector, positive      1
Vector, support of      1
Vertex      30
Vertex of a digraph      38
Vertex, adjacent      31
Vertex, degree      31
Vertex, incident      31
Vertex, neighbor      31
Vertex, pendant      31
Vertices, connected      32
Walk      32
Walk, closed      32
Walk, length of      32
Walk, open      32
Weak perfect graph conjecture      41

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