A norm inequality for positive block matrices

Minghua Lin

doi:10.1016/j.crma.2018.05.006

Functional analysis

A norm inequality for positive block matrices
[Une inégalité de norme pour les matrices positives écrites par blocs]

Présenté par : the Editorial Board

Minghua Lin ¹

¹ Department of Mathematics, Shanghai University, Shanghai, 200444, China

Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 818-822.

Résumé
Abstract

Toute matrice positive $M = {(M_{i, j})}_{i, j = 1}^{m}$ écrite en blocs carrés $M_{i, j}$ satisfait $‖ M ‖ \leq ‖ {\sum_{i = 1}}^{m} M_{i, i} + \sum_{i = 1}^{m - 1} ω_{i} I ‖$ , où les quantités $ω_{i}$ , $i = 1, \dots, m - 1$ , font intervenir la largeur du domaine des valeurs numériques. Ceci étend le théorème principal de Bourin, Mhanna (2017) [4] aux matrices écrites avec un nombre de blocs arbitraire.

Any positive matrix $M = {(M_{i, j})}_{i, j = 1}^{m}$ with each block $M_{i, j}$ square satisfies the symmetric norm inequality $‖ M ‖ \leq ‖ \sum_{i = 1}^{m} M_{i, i} + \sum_{i = 1}^{m - 1} ω_{i} I ‖$ , where $ω_{i}$ ( $i = 1, \dots, m - 1$ ) are quantities involving the width of numerical ranges. This extends the main theorem of Bourin and Mhanna (2017) [4] to a higher number of blocks.

Reçu le : 2018-05-02
Accepté le : 2018-05-15
Publié le : 2018-05-21

DOI : 10.1016/j.crma.2018.05.006

Affiliations des auteurs :

Minghua Lin ¹

¹ Department of Mathematics, Shanghai University, Shanghai, 200444, China

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     author = {Minghua Lin},
     title = {A norm inequality for positive block matrices},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {818--822},
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     number = {7},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.006},
     language = {en},
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Minghua Lin. A norm inequality for positive block matrices. Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 818-822. doi : 10.1016/j.crma.2018.05.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.05.006/

Bibliographie
Cité par

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[4] J.-C. Bourin; A. Mhanna Positive block matrices and numerical ranges, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017), pp. 1077-1081

[5] M. Gumus; J. Liu; S. Raouafi; T.-Y. Tam Positive semi-definite $2 \times 2$ block matrices and norm inequalities, Linear Algebra Appl., Volume 551 (2018), pp. 83-91

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[8] A. Mhanna On symmetric norm inequalities and positive definite block-matrices, Math. Inequal. Appl., Volume 21 (2018), pp. 133-138

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