Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices

Jean-Christophe Bourin; Jingjing Shao

doi:10.5802/crmath.25

Analyse fonctionnelle

Convex maps on

ℝ^{n}

and positive definite matrices

Jean-Christophe Bourin ¹ ; Jingjing Shao ²

¹ Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
² College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China

Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 645-649.

Résumé

We obtain several convexity statements involving positive definite matrices. In particular, if $A, B, X, Y$ are invertible matrices and $A, B$ are positive, we show that the map

(s, t) \mapsto Tr log (X^{*} A^{s} X + Y^{*} B^{t} Y)

is jointly convex on $ℝ^{2}$ . This is related to some exotic matrix Hölder inequalities such as

∥sinh (\sum_{i = 1}^{m} A_{i} B_{i})∥ \leq {∥sinh (\sum_{i = 1}^{m} A_{i}^{p})∥}^{1 / p} {∥sinh (\sum_{i = 1}^{m} B_{i}^{q})∥}^{1 / q}

for all positive matrices $A_{i}, B_{i}$ , such that $A_{i} B_{i} = B_{i} A_{i}$ , conjugate exponents $p, q$ and unitarily invariant norms $∥ \cdot ∥$ . Our approach to obtain these results consists in studying the behaviour of some functionals along the geodesics of the Riemanian manifold of positive definite matrices.

Reçu le : 2019-09-24
Révisé le : 2020-02-13
Accepté le : 2020-06-15
Publié le : 2020-10-02

DOI : 10.5802/crmath.25

Classification : 47A30, 15A60

Affiliations des auteurs :

Jean-Christophe Bourin ¹ ; Jingjing Shao ²

¹ Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
² College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_6_645_0,
     author = {Jean-Christophe Bourin and Jingjing Shao},
     title = {Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {645--649},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.25},
     language = {en},
}

TY  - JOUR
AU  - Jean-Christophe Bourin
AU  - Jingjing Shao
TI  - Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 645
EP  - 649
VL  - 358
IS  - 6
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.25
LA  - en
ID  - CRMATH_2020__358_6_645_0
ER  -

%0 Journal Article
%A Jean-Christophe Bourin
%A Jingjing Shao
%T Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices
%J Comptes Rendus. Mathématique
%D 2020
%P 645-649
%V 358
%N 6
%I Académie des sciences, Paris
%R 10.5802/crmath.25
%G en
%F CRMATH_2020__358_6_645_0

Jean-Christophe Bourin; Jingjing Shao. Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 645-649. doi : 10.5802/crmath.25. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/

Bibliographie
Cité par

[1] T. Ando Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., Volume 26 (1979), pp. 203-241 | DOI | MR | Zbl

[2] Rajendra Bhatia Matrix Analysis, Graduate Texts in Mathematics, 169, Springer, 1996 | Zbl

[3] Rajendra Bhatia Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, 2007 | Zbl

[4] Jean-Christophe Bourin; Eun-Young Lee Matrix inequalities from a two variables functional, Int. J. Math., Volume 27 (2016) no. 9, 16500771, 19 pages | MR | Zbl

[5] Constantin P. Niculescu Convexity according to the geometric mean, Math. Inequal. Appl., Volume 3 (2000) no. 2, pp. 155-167 | MR | Zbl

Saja Hayajneh; Fuad Kittaneh A log-majorization version of Audenaert's inequality, Journal of Mathematical Analysis and Applications, Volume 548 (2025) no. 1, p. 129372 | DOI:10.1016/j.jmaa.2025.129372
Fumio Hiai Log-majorization and matrix norm inequalities with application to quantum information, Acta Scientiarum Mathematicarum, Volume 90 (2024) no. 3-4, p. 527 | DOI:10.1007/s44146-024-00142-w

Cité par 2 documents. Sources : Crossref

Commentaires - Politique