Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices Antoine Mhanna To cite this version: Antoine Mhanna. Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices. 2015. hal-01182244v5 HAL Id: hal-01182244 https://hal.inria.fr/hal-01182244v5 Preprint submitted on 14 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices Antoine Mhanna1 1 Dept of Mathematics, Lebanese University, Hadath, Beirut, Lebanon.
[email protected] Abstract For positive semi-definite block-matrix M, we say that M is P.S.D. and we write A X M = M+ , with A M+, B M+ . The focus is on studying the conse- X B ∈ n+m ∈ n ∈ m ∗ quences of a decomposition lemma due to C. Bourrin and the main result is extending the class of P.S.D. matrices M written by blocks of same size that satisfies the inequality: M A + B for all symmetric norms. k k ≤ k k Keywords : Matrix Analysis, Hermitian matrices, symmetric norms. 1 Introduction Let A be an n n matrix and F an m m matrix, (m>n) written by blocks such that A × × is a diagonal block and all entries other than those of A are zeros, then the two matrices have the same singular values and for all unitarily invariant norms A = F = A 0 , k k k k k ⊕ k we say then that the symmetric norm on Mm induces a symmetric norm on Mn, so for square matrices we may assume that our norms are defined on all spaces Mn, n 1.