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Keywords:
completely positive/copositive matrix; proper cone; semipositive matrix; positive semidefinite matrix; linear preserver problem
completely positive/copositive matrix; proper cone; semipositive matrix; positive semidefinite matrix; linear preserver problem
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 $\mathcal {S}^2_+$.
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