# 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

Research output: Contribution to journalArticleAcademicpeer-review

9 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 language Undefined 191-206 11 Linear algebra and its applications 482 https://doi.org/10.1016/j.laa.2015.05.021 Published - 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