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

TL;DR

This paper characterizes Hermitian matrices with variable diagonals that minimize the operator norm quotient, linking their properties to positive matrices and majorization theory, and provides a constructive method for their generation.

Contribution

It introduces a new characterization of minimal Hermitian matrices with variable diagonals and develops a constructive approach based on majorization theory.

Findings

Hermitian matrices with minimal quotient norm are related to specific positive matrices.

A constructive method for obtaining minimal matrices of any dimension is proposed.

The problem is connected to majorization results in R^n.

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 R^n.