[Sur les transformations positives des matrices d'une dimension donnée]
Un résultat classique de I.J. Schoenberg caractérise les fonctions préservant la positivité lorsqu'elles sont appliquées aux entrées des matrices semi-définies positives de dimension arbitraire. Le problème analogue lorsque la dimension est fixe est beaucoup plus complexe à résoudre. Dans cette note, nous résolvons ce problème dans le cas où la fonction est un polynôme. Nous dérivons de ce résultat des inégalités exactes pour les puissances d'Hadamard d'une matrice positive et pour le problème du cube matriciel.
A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomial case. As consequences, we derive tight linear matrix inequalities for Hadamard powers of positive semidefinite matrices, and a sharp asymptotic bound for the matrix cube problem involving Hadamard powers.
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@article{CRMATH_2016__354_2_143_0,
author = {Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai},
title = {Matrix positivity preservers in fixed dimension},
journal = {Comptes Rendus. Math\'ematique},
pages = {143--148},
publisher = {Elsevier},
volume = {354},
number = {2},
year = {2016},
doi = {10.1016/j.crma.2015.11.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.11.006/}
}
TY - JOUR AU - Belton, Alexander AU - Guillot, Dominique AU - Khare, Apoorva AU - Putinar, Mihai TI - Matrix positivity preservers in fixed dimension JO - Comptes Rendus. Mathématique PY - 2016 SP - 143 EP - 148 VL - 354 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.11.006/ DO - 10.1016/j.crma.2015.11.006 LA - en ID - CRMATH_2016__354_2_143_0 ER -
%0 Journal Article %A Belton, Alexander %A Guillot, Dominique %A Khare, Apoorva %A Putinar, Mihai %T Matrix positivity preservers in fixed dimension %J Comptes Rendus. Mathématique %D 2016 %P 143-148 %V 354 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.11.006/ %R 10.1016/j.crma.2015.11.006 %G en %F CRMATH_2016__354_2_143_0
Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai. Matrix positivity preservers in fixed dimension. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 143-148. doi : 10.1016/j.crma.2015.11.006. http://www.numdam.org/articles/10.1016/j.crma.2015.11.006/
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