Linear mappings preserving the copositive cone

Shitov, Yaroslav

doi:10.1090/proc/15432

Linear mappings preserving the copositive cone
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by Yaroslav Shitov

Proc. Amer. Math. Soc. 149 (2021), 3173-3176

DOI: https://doi.org/10.1090/proc/15432

Published electronically: May 11, 2021

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Abstract:

Let $\mathcal {S}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let $\mathcal {C}_n$ be the copositive cone, that is, the set of all matrices $a\in \mathcal {S}_n$ that fulfill the condition $u^\top a u\geqslant 0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $\varphi :\mathcal {S}_n\to \mathcal {S}_n$ satisfies $\varphi (\mathcal {C}_n)=\mathcal {C}_n$ if and only if \begin{equation*} \varphi (x)=m^\top xm \end{equation*} for a fixed monomial matrix $m$ with nonnegative entries.

References

Abraham Berman, Mirjam Dür, and Naomi Shaked-Monderer, Open problems in the theory of completely positive and copositive matrices, Electron. J. Linear Algebra 29 (2015), 46–58. MR 3414584, DOI 10.13001/1081-3810.2943
Peter J. C. Dickinson and Luuk Gijben, On the computational complexity of membership problems for the completely positive cone and its dual, Comput. Optim. Appl. 57 (2014), no. 2, 403–415. MR 3165055, DOI 10.1007/s10589-013-9594-z
G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. Berlin (1897) 994–1015.
S. Furtado, C. R. Johnson, and Y. Zhang, Linear preservers of copositive matrices, Linear Multilinear A. (2019) doi.org/10.1080/03081087.2019.1692775.
Alexander Guterman, Chi-Kwong Li, and Peter emrl, Some general techniques on linear preserver problems, Linear Algebra Appl. 315 (2000), no. 1-3, 61–81. MR 1774960, DOI 10.1016/S0024-3795(00)00119-1
Marshall Hall Jr., Combinatorial theory, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0224481
J.-B. Hiriart-Urruty and A. Seeger, A variational approach to copositive matrices, SIAM Rev. 52 (2010), no. 4, 593–629. MR 2736966, DOI 10.1137/090750391
Chi-Kwong Li and Stephen Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), no. 7, 591–605. MR 1862098, DOI 10.2307/2695268
Chi-Kwong Li and Nam-Kiu Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162/164 (1992), 217–235. Directions in matrix theory (Auburn, AL, 1990). MR 1148401, DOI 10.1016/0024-3795(92)90377-M
John E. Maxfield and Henryk Minc, On the matrix equation $X^{\prime } X=A$, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 125–129. MR 174570, DOI 10.1017/S0013091500014681
Hannu Väliaho, Criteria for copositive matrices, Linear Algebra Appl. 81 (1986), 19–34. MR 852891, DOI 10.1016/0024-3795(86)90246-6