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 | English |
---|---|
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
- n/a OA procedure