Completely Positive Matrices
A real matrix is positive semidefinite if it can be decomposed as A = BBOC . 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 = BBOC 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: Preliminaries: 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?. Readership: Upper level undergraduates, graduate students, academics and researchers interested in matrix theory."
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adjacent assume Berman block form Cauchy matrix chordal chordal graphs closed convex cone complete graph completely positive graph completely positive matrix connected graph convex cone copositive Corollary CP matrix realization cp-rank G cut vertex denote diagonal entries diagonal matrix diagonally dominant DNN matrix realization doubly nonnegative matrix eigenvalues equal erists Example Exercise extreme rays G1 and G2 Gram matrix graph G implies irreducible Lemma M-matrix n x n completely positive n x n matrix NCC graph nonnegative vector nonsingular nonzero odd cycle partial CP partial symmetric matrix pletely positive positive definite positive diagonal positive semidefinite matrix principal minors principal submatrix proof of Theorem property PLSS Prove Proposition PSD completion PSD matrix rank 1 representation rank A1 realization of G satisfies 3.30 Schur complement singular subgraph of G Suppose symmetric matrix totally nonnegative matrices triangle free graph UL-completely positive