Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices
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 \[ (s,t) \mapsto \mathrm{Tr}\,\log \left(X^*A^sX + Y^*B^tY\right) \] is jointly convex on $\mathbb{R}^2$. This is related to s...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2020-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/ |
| _version_ | 1827785280799637504 |
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| author | Bourin, Jean-Christophe Shao, Jingjing |
| author_facet | Bourin, Jean-Christophe Shao, Jingjing |
| author_sort | Bourin, Jean-Christophe |
| collection | DOAJ |
| description | 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
\[ (s,t) \mapsto \mathrm{Tr}\,\log \left(X^*A^sX + Y^*B^tY\right) \]
is jointly convex on $\mathbb{R}^2$. This is related to some exotic matrix Hölder inequalities such as
\[ \left\Vert \sinh \left(\sum _{i=1}^m A_iB_i\right) \right\Vert \le \left\Vert \sinh \left(\sum _{i=1}^m A_i^p\right) \right\Vert ^{1/p} \left\Vert \sinh \left(\sum _{i=1}^m B_i^q\right) \right\Vert ^{1/q} \]
for all positive matrices $A_i, B_i$, such that $A_iB_i=B_iA_i$, conjugate exponents $p,q$ and unitarily invariant norms $\Vert \cdot \Vert $. 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. |
| first_indexed | 2024-03-11T16:16:56Z |
| format | Article |
| id | doaj.art-1490b99409354c81835a41ee1af48ee0 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:56Z |
| publishDate | 2020-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-1490b99409354c81835a41ee1af48ee02023-10-24T14:18:57ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692020-10-01358664564910.5802/crmath.2510.5802/crmath.25Convex maps on $\protect \mathbb{R}^n$ and positive definite matricesBourin, Jean-Christophe0Shao, Jingjing1Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, FranceCollege of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, ChinaWe 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 \[ (s,t) \mapsto \mathrm{Tr}\,\log \left(X^*A^sX + Y^*B^tY\right) \] is jointly convex on $\mathbb{R}^2$. This is related to some exotic matrix Hölder inequalities such as \[ \left\Vert \sinh \left(\sum _{i=1}^m A_iB_i\right) \right\Vert \le \left\Vert \sinh \left(\sum _{i=1}^m A_i^p\right) \right\Vert ^{1/p} \left\Vert \sinh \left(\sum _{i=1}^m B_i^q\right) \right\Vert ^{1/q} \] for all positive matrices $A_i, B_i$, such that $A_iB_i=B_iA_i$, conjugate exponents $p,q$ and unitarily invariant norms $\Vert \cdot \Vert $. 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/ |
| spellingShingle | Bourin, Jean-Christophe Shao, Jingjing Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices Comptes Rendus. Mathématique |
| title | Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices |
| title_full | Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices |
| title_fullStr | Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices |
| title_full_unstemmed | Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices |
| title_short | Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices |
| title_sort | convex maps on protect mathbb r n and positive definite matrices |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/ |
| work_keys_str_mv | AT bourinjeanchristophe convexmapsonprotectmathbbrnandpositivedefinitematrices AT shaojingjing convexmapsonprotectmathbbrnandpositivedefinitematrices |