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基本説明
Focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp rank.
Full Description
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.
Contents
Matrix Theoretic Background; Positive Semidefinite Matrices; Nonnegative Matrices and M-Matrices; Schur Complements; Graphs; Convex Cones; The PSD Completion Problem; Complete Positivity: Definition and Basic Properties; Cones of Completely Positive Matrices; Small Matrices; Complete Positivity and the Comparison Matrix; Completely Positive Graphs; Completely Positive Matrices Whose Graphs are Not Completely Positive; Square Factorizations; Functions of Completely Positive Matrices; The CP Completion Problem; CP Rank: Definition and Basic Results; Completely Positive Matrices of a Given Rank; Completely Positive Matrices of a Given Order; When is the CP-Rank Equal to the Rank?
