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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2018-05-02
Accepted: 2018-05-15
Published online: 2018-05-21

DOI: 10.1016/j.crma.2018.05.006

Author's affiliations:

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},
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     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/

References
Cited by

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[2] J.-C. Bourin; E.-Y. Lee Decomposition and partial trace of positive matrices with Hermitian blocks, Int. J. Math., Volume 24 (2013)

[3] J.-C. Bourin; E.-Y. Lee; M. Lin On a decomposition lemma for positive semidefinite block matrices, Linear Algebra Appl., Volume 437 (2012), pp. 1906-1912

[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

[6] T. Hiroshima Majorization criterion for distillability of a bipartite quantum state, Phys. Rev. Lett., Volume 91 (2003) no. 5

[7] R.A. Horn; C.R. Johnson Topics in Matrix Analysis, Cambridge University Press, 1991

[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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