The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Peter James Clair Dickinson, Immanuel M. Bomze, Georg J. Still

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6 Citations (Scopus)

Abstract

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.
Original languageUndefined
Pages (from-to)191-206
Number of pages11
JournalLinear algebra and its applications
Volume482
DOIs
Publication statusPublished - 1 Oct 2015

Keywords

  • EWI-26090
  • MSC-90C25
  • MSC-15A23
  • MSC-15B48
  • Nonnegative factorisation
  • CP-rank
  • Completely positive matrices
  • Genericity
  • Copositive optimisation
  • METIS-312644
  • IR-96230

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