On linear maps leaving invariant the copositive/completely positive cones

Jayaraman, Sachindranath, and Mer, Vatsalkumar N.. "On linear maps leaving invariant the copositive/completely positive cones." Czechoslovak Mathematical Journal 74.3 (2024): 801-815. <http://eudml.org/doc/299313>.

@article{Jayaraman2024,
abstract = {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 $n \le 4$, 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 $n \le 4$) 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 $\mathcal \{S\}^2_+$.},
author = {Jayaraman, Sachindranath, Mer, Vatsalkumar N.},
journal = {Czechoslovak Mathematical Journal},
keywords = {completely positive/copositive matrix; proper cone; semipositive matrix; positive semidefinite matrix; linear preserver problem},
language = {eng},
number = {3},
pages = {801-815},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On linear maps leaving invariant the copositive/completely positive cones},
url = {http://eudml.org/doc/299313},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Jayaraman, Sachindranath
AU - Mer, Vatsalkumar N.
TI - On linear maps leaving invariant the copositive/completely positive cones
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 801
EP - 815
AB - 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 $n \le 4$, 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 $n \le 4$) 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 $\mathcal {S}^2_+$.
LA - eng
KW - completely positive/copositive matrix; proper cone; semipositive matrix; positive semidefinite matrix; linear preserver problem
UR - http://eudml.org/doc/299313
ER -