Completely Positive Matrices (Google eBook)
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 work 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.
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A/-matrix adjacent assume Barioli Berman block form Cauchy matrix chordal graphs closed convex cone complete graph completely positive graph completely positive matrix connected graph convex cone COPn copositive Corollary CP matrix realization cp-rank G cut vertex cycle of length denote diagonal entries diagonal matrix diagonally dominant DNN matrix realization doubly nonnegative matrix eigenvalues equal Example Exercise exists extreme rays G-partial Gram matrix graph G implies irreducible Lemma M-matrix matrix whose graph n x n completely positive n x n doubly nonnegative n x n matrix NCC graph nonnegative vector nonsingular nonzero 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 realization of G satisfies 3.30 Schur complement singular subgraph of G subspace suppb Suppose symmetric matrix symmetric nonnegative totally nonnegative matrix triangle free graph