On the numerical range of some block matrices with scalar diagonal blocks

TL;DR

This paper establishes new verifiable conditions under which certain block matrices with scalar diagonal blocks have their numerical range as the convex hull of a limited number of ellipses, with necessary conditions for specific cases.

Contribution

It introduces new conditions for the numerical range of block matrices with scalar diagonal blocks, including necessary conditions for the case when the block size is two.

Findings

Conditions for the numerical range to be the convex hull of at most k ellipses

Necessary conditions for the case k=2 with co-centered ellipses

Extension of previous results to broader classes of block matrices

Abstract

Several new verifiable conditions are established for block matrices with scalar diagonal blocks to have the numerical range equal the convex hull of at most k ellipses where k by k is the size of the smaller diagonal block. For k = 2, these conditions are also necessary, provided that the ellipses are co-centered.