Completely Positive MatricesWorld Scientific, 2003 - 206 Seiten 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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A₁ adjacent assume b₁ Berman block form c₁ Cauchy matrix chordal chordal graphs closed convex cone complete graph completely positive graph completely positive matrix convex cone COPn copositive Corollary CP matrix realization cp-rank G cut vertex denote diagonal entries diagonal matrix diagonally dominant DNN matrix realization doubly nonnegative matrix eigenvalues Example Exercise exists extreme rays G₁ and G2 Gram matrix graph G implies irreducible K₁ Lemma M-matrix matrix whose graph n x n completely positive n x n matrix nonnegative vector nonsingular odd cycle partial symmetric matrix permutation matrix pletely positive positive definite positive semidefinite matrix principal minors principal submatrix proof of Theorem property PLSS Prove Proposition PSD completion rank 1 representation rank A1 realization of G satisfies 3.30 Schur complement singular subgraph of G supp Suppose symmetric matrix totally nonnegative matrix triangle free graph UL-completely positive Ω₁