The normalized numerical range and the Davis-Wielandt shell

TL;DR

This paper characterizes the normalized numerical range of matrices, especially normal and 2x2 matrices, by linking it to the Davis-Wielandt shell through a novel nonlinear mapping, extending previous results.

Contribution

It provides an explicit description of the normalized numerical range for normal and 2x2 matrices using the Davis-Wielandt shell framework.

Findings

Explicit description for normal matrices

Explicit description for 2x2 matrices

New perspective via the Davis-Wielandt shell

Abstract

For a given $n$-by-$n$ matrix $A$, its {\em normalized numerical range} $F_N(A)$ is defined as the range of the function $f_{N,A}\colon x\mapsto (x^*Ax)/(\norm{Ax}\cdot\norm{x})$ on the complement of $\ker A$. We provide an explicit description of this set for the case when $A$ is normal or $n=2$. This extension of earlier results for particular cases of $2$-by-$2$ matrices (by Gevorgyan) and essentially Hermitian matrices of arbitrary size (by A. Stoica and one of the authors) was achieved due to the fresh point of view at $F_N(A)$ as the image of the Davis-Wielandt shell $\JNR(A)$ under a certain non-linear mapping $h\colon\R^3\mapsto\C$.