Matrix Positivity

Copyright
Contents
Preface
List of Symbols
1 Background
	1.1 Purpose
	1.2 Matrices
	1.3 Convexity
	1.4 Theorem of the Alternative
2 Positivity Classes
	2.1 Introduction
	2.2 (Entry-wise) Positive Matrices (EP)
	2.3 M-Matrices (M)
	2.4 Totally Positive Matrices (TP)
	2.5 Positive Definite Matrices (PD)
	2.6 Strictly Copositive Matrices (SC)
	2.7 Doubly Nonnegative Matrices (DN)
	2.8 Completely Positive Matrices (CP)
	2.9 P-Matrices (P)
	2.10 H+-Matrices (H+)
	2.11 Monotone Matrices (Mon)
	2.12 Semipositive Matrices (SP)
	2.13 Relationships among the Classes
3 Semipositive Matrices
	3.1 Definitions and Elementary Properties
	3.2 Sumsand Products of SP Matrices
	3.3 Further Structure and Sign Patterns of SP Matrices
	3.4 Spectral Theoryof SP Matrices
	3.5 Minimally Semipositive Matrices
	3.6 Linear Preservers
	3.7 Geometric Mapping Properties of SP Matrices
	3.8 Strictly Semimonotone Matrices
4 P-Matrices
	4.1 Introduction
	4.2 Notation, Definitions, and Preliminaries
	4.3 Basic Properties of P-Matrices
	4.4 The Eigenvalues of P-Matrices
	4.5 Special Classes of P-Matrices
	4.6 The P-Problem: Detection of P-Matrices
	4.7 The Recursive Construction of All P-Matrices
	4.8 On Matrices with Nonnegative Principal Minors
	4.9 Some Applications of P-Matrices
	4.10 Miscellaneous Related Facts and References
5 Inverse M-Matrices
	5.1 Introduction
	5.2 Preliminary Facts
	5.3 Diagonal Closure
	5.4 Diagonal Lyapunov Solutions
	5.5 Additive Diagonal Closure
	5.6 Power Invariant Zero Pattern
	5.7 Sufficient Conditions for a Positive Matrix to Be IM
	5.8 Inverse M-Matrices Have Roots in IM
	5.9 Partitioned IM Matrices
	5.10 Submatrices
	5.11 Vanishing Almost Principal Minors
	5.12 The Path Product Property
	5.13 Triangular Factorizations
	5.14 Sums, Products, and Closure
	5.15 Spectral Structure of IM Matrices
	5.16 Hadamard Products and Powers, Eventually IM Matrices
	5.17 Perturbation of IM Matrices
	5.18 Determinantal Inequalities
	5.19 Completion Theory
	5.20 Connections with Other Matrix Classes
	5.21 Graph Theoretic Characterizations of IM Matrices
	5.22 Linear Interpolation Problems
	5.23 Newton Matrices
	5.24 Perron Complements of IM Matrices
	5.25 Products of IM Matrices
	5.26 Topological Closure of IM Matrices
	5.27 Tridiagonal, Triangular, and Reducible IM Matrices
	5.28 Ultrametric Matrices
6 Copositive Matrices
	6.1 Introduction
	6.2 Basic Properties of Copositive Matrices in Mn(R)
	6.3 Characterizations of C, SC, and C+ Matrices
	6.4 History
	6.5 Spectral Properties
	6.6 Linear Preservers
References
Index