Comptes Rendus
no. 2
Algebra/Mathematical analysis
Matrix positivity preservers in fixed dimension
[Sur les transformations positives des matrices d'une dimension donnée]

Présenté par : the Editorial Board

Alexander Belton1 ; Dominique Guillot2 ; Apoorva Khare3 ; Mihai Putinar4, 5
1 Lancaster University, Lancaster, UK
2 University of Delaware, Newark, DE, USA
3 Stanford University, Stanford, CA, USA
4 University of California at Santa Barbara, CA, USA
5 Newcastle University, Newcastle upon Tyne, UK
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 143-148.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.11.006
Alexander Belton 1 ; Dominique Guillot 2 ; Apoorva Khare 3 ; Mihai Putinar 4, 5

1 Lancaster University, Lancaster, UK
2 University of Delaware, Newark, DE, USA
3 Stanford University, Stanford, CA, USA
4 University of California at Santa Barbara, CA, USA
5 Newcastle University, Newcastle upon Tyne, UK 
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Alexander Belton; Dominique Guillot; Apoorva Khare; Mihai Putinar. Matrix positivity preservers in fixed dimension. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 143-148. doi : 10.1016/j.crma.2015.11.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.006/

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