This book is an updated and extended version of Completely Positive Matrices (Abraham Berman and Naomi Shaked-Monderer, World Scientific 2003). It contains new sections on the cone of copositive matrices, which is the dual of the cone of completely positive matrices, and new results on both copositive matrices and completely positive matrices.The book is an up to date comprehensive resource for researchers in Matrix Theory and Optimization. It can also serve as a textbook for an advanced undergraduate or graduate course.
Author(s): Naom Shaked-Monderer, Abraham Berman
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 561
City: Singapore
Contents
Preface
1. Background
1.1 Matrix theoretic background
1.2 Positive semidefinite matrices
1.3 Nonnegative matrices and M-matrice
1.4 Schur complements
1.5 Graphs
1.6 Convex cones
1.7 Optimization and the Karush-Kuhn-Tucker conditions
1.8 The PSD completion problem
2. Copositivity
2.1 Definition and basic properties
2.2 Spectral properties of copositive matrices
2.3 Cones of copositive matrices
2.4 Zeros of copositive matrices
2.5 M-irreducibility, exceptionality and extremality
2.6 Methods for determining copositivity
2.7 Almost copositive matrices
2.8 Copositive {0, 1, –1}-matrices and related matrices
2.9 Small copositive matrices
2.10 COP5
2.11 Exceptional extremal copositive matrices
2.12 The inverse of a copositive matrix
2.13 The COP and SPN completion problems
2.14 SPN graphs
3. Complete positivity
3.1 Definition and basic properties
3.2 Cones of completely positive matrices
3.3 Small completely positive matrices
3.4 Complete positivity and the comparison matrix
3.5 Completely positive graphs
3.6 Completely positive matrices whose graphs are not completely positive
3.7 CP5
3.8 Square and rank-revealing CP factorizations
3.9 Functions of completely positive matrices
3.10 The CP completion problem
3.11 Rational and integral completely positive matrices
4. CP-Rank
4.1 Definition and basic results
4.2 Completely positive matrices of a given rank
4.3 The cp-ranks and minimal CP factorizations in CPn
4.4 Completely positive matrices of a given order, with a given graph
4.5 Bounding pn
4.6 When is the cp-rank equal to the rank?
4.7 Graphs attaining minimal cp-rank
4.8 The number of (minimal) CP factorizations
4.9 Rational and integral cp-rank
5. The structure of COPn and CPn
5.1 The structure of the copositive cone
5.2 The structure of CPn
Bibliography
Notation
Index
