New pairs of matrices with convex generalized numerical ranges

TL;DR

This paper investigates conditions under which the generalized numerical range of certain block matrices remains convex, focusing on matrices formed by direct sums with identity matrices and their 2x2 components.

Contribution

It identifies specific matrix structures involving 2x2 blocks and identities that preserve convexity of the generalized numerical range.

Findings

Convexity of the generalized numerical range is maintained for matrices of the form A=hatA⊕Ik, B=hatB⊕Ik.

The generalized numerical range W_A(B) equals W_{hatA}(hatB) under certain block matrix conditions.

The study provides new matrix pairs with convex generalized numerical ranges.

Abstract

In this article, we are going to search for $n\times n$ matrices $A$ and $B$ such that their generalized numerical range $$W_A(B)=\{tr(AU^*BU) \ :\ U^*U=UU^*=I\}$$ is convex. More specifically, we consider $A=\hat{A}\oplus I_k$ and $B=\hat{B}\oplus I_k$ where $\hat{A}$ and $\hat{B}$ are $2\times 2$. If $W_A(B)=W_{\hat{A}}(\hat{B})$ then it is a convex set.