A characterization of Hermitian matrices with variable diagonal and smallest operator norm

2011/04/19 by Andruchow, Esteban, Larotonda, Gabriel, Recht, Lázaro +1
#15A12 #15A60 #15B51 #15B57 #58B25 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1104.3841

Abstract

We describe properties of a Hermitian square matrix M in M_n(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in M_n(C) and || || the operator norm. These matrices are related to some particular positive matrices with their range included in the eigenspaces of the eigenvalues +||M|| and -||M|| of M. We show how a constructive method can be used to obtain minimal matrices of any dimension relating this problem with majorization results in Rⁿ.

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