The possible shapes of numerical ranges

TL;DR

This paper characterizes all convex sets in the complex plane that can be realized as the numerical range of some matrix, and shows any matrix's numerical range can be matched by a symmetric matrix.

Contribution

It provides a complete characterization of convex numerical ranges and proves the existence of symmetric matrices with the same numerical range as any given matrix.

Findings

Characterization of convex sets as numerical ranges

Existence of symmetric matrices with identical numerical ranges

Extension of numerical range properties to symmetric matrices

Abstract

Which convex subsets of the complex plane are the numerical range W(A of some matrix A? This paper gives a precise characterization of these sets. In addition to this we show that for any A there exists a symmetric matrix B of the same size such that W(A)=W(B).