A characterization of Hermitian matrices with variable diagonal and smallest operator norm
Abstract
We describe properties of a Hermitian square matrix M in Mn(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in Mn(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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