Article
| Title: | On linear maps leaving invariant the copositive/completely positive cones (English) |
| Author: | Jayaraman, Sachindranath |
| Author: | Mer, Vatsalkumar N. |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 3 |
| Year: | 2024 |
| Pages: | 801-815 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices $\mathcal {S}^n$ that leave invariant the closed convex cones of copositive and completely positive matrices (${\rm COP}_n$ and ${\rm CP}_n$). A description of an invertible linear map on $\mathcal {S}^n$ such that $L({\rm CP}_n) \subset CP_n$ is obtained in terms of semipositive maps over the positive semidefinite cone $\mathcal {S}^n_+$ and the cone of symmetric nonnegative matrices $\mathcal {N}^n_+$ for $n \leq 4$, with specific calculations for $n=2$. Preserver properties of the Lyapunov map $X \mapsto AX + XA^t$, the generalized Lyapunov map $X \mapsto AXB + B^tXA^t$, and the structure of the dual of the cone $\pi ({\rm CP} _n)$ (for $n \leq 4$) are brought out. We also highlight a different way to determine the structure of an invertible linear map on $\mathcal {S}^2$ that leaves invariant the closed convex cone $\mathcal {S}^2_+$. (English) |
| Keyword: | completely positive/copositive matrix |
| Keyword: | proper cone |
| Keyword: | semipositive matrix |
| Keyword: | positive semidefinite matrix |
| Keyword: | linear preserver problem |
| MSC: | 15A86 |
| MSC: | 15B48 |
| DOI: | 10.21136/CMJ.2024.0002-24 |
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| Date available: | 2024-10-03T12:37:13Z |
| Last updated: | 2024-10-04 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152582 |
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