The structure of completely positive matrices according to their CP-rank and CP-plus-rank
| Autor: | Peter J. C. Dickinson, Georg Still, Immanuel M. Bomze |
|---|---|
| Přispěvatelé: | Discrete Mathematics and Mathematical Programming |
| Rok vydání: | 2015 |
| Předmět: |
Discrete mathematics
Numerical Analysis Algebra and Number Theory MSC-15A23 Generic property CP-rank Operator (physics) MSC-90C25 Completely positive matrices Structure (category theory) MSC-15B48 Copositive optimisation Genericity Combinatorics EWI-26090 Nonnegative factorisation Discrete Mathematics and Combinatorics Rank (graph theory) Geometry and Topology IR-96230 METIS-312644 Mathematics |
| Zdroj: | Linear algebra and its applications, 482, 191-206. Elsevier |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2015.05.021 |
| Popis: | We study the topological properties of the cp-rank operator $\mathrm{cp}(A)$ and the related cp-plus-rank operator $\mathrm{cp}^+(A)$ (which is introduced in this paper) in the set $\mathcal{S}^n$ of symmetric $n\times n$-matrices. For the set of completely positive matrices, $\mathcal{CP}^n$, we show that for any fixed p the set of matrices A satisfying $\mathrm{cp}(A)=\mathrm{cp}^+(A)=p$ is open in $\mathcal{S}^n\setminus\mathrm{bd}(\mathcal{CP}^n)$. We also prove that the set $\mathcal{A}^n$ of matrices with $\mathrm{cp}(A)=\mathrm{cp}^+(A)$ is dense in $\mathcal{S}^n$. By applying the theory of semi-algebraic sets we are able to show that membership in $\mathcal{A}^n$ is even a generic property. We furthermore answer several questions on the existence of matrices satisfying $\mathrm{cp}(A)=\mathrm{cp}^+(A)$ or $\mathrm{cp}(A)\neq\mathrm{cp}^+(A)$, and establish genericity of having infinitely many minimal cp-decompositions. |
| Databáze: | OpenAIRE |
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