### 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 |
---|---|

Pages (from-to) | 191-206 |

Number of pages | 11 |

Journal | Linear algebra and its applications |

Volume | 482 |

DOIs | |

Publication status | 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

## Cite this

Dickinson, P. J. C., Bomze, I. M., & Still, G. J. (2015). The structure of completely positive matrices according to their CP-rank and CP-plus-rank.

*Linear algebra and its applications*,*482*, 191-206. https://doi.org/10.1016/j.laa.2015.05.021