Infimum of a matrix norm of A induced by an absolute vector norm

TL;DR

This paper characterizes the smallest possible matrix norm induced by an absolute vector norm for a square matrix, revealing conditions when it equals the spectral radius, especially for sign equivalent matrices.

Contribution

It provides a characterization of the infimum of induced matrix norms over real and complex fields, linking it to spectral properties and sign equivalence.

Findings

Infimum of the induced matrix norm can be greater than the spectral radius.

If A is sign equivalent to a nonnegative matrix B, the infimum equals the spectral radius of B.

The characterization applies over both real and complex fields.

Abstract

We characterize the infimum of a matrix norm of a square matrix A induced by an absolute norm, over the fields of real and complex numbers. Usually this infimum is greater than the spectral radius of A. If A is sign equivalent to a nonnegative matrix B then this infimum is the spectral radius of B.