Higher rank numerical ranges of normal matrices

TL;DR

This paper characterizes the higher rank numerical ranges of normal matrices, showing they can be represented as intersections of a limited number of half planes and providing constructions for matrices with prescribed numerical ranges.

Contribution

It establishes bounds on the number of half planes needed to describe higher rank numerical ranges of normal matrices and constructs matrices with specified convex polygonal ranges.

Findings

Higher rank numerical range of a normal matrix can be expressed as intersection of at most max{m,4} half planes.

Constructs normal matrices with minimal size for a given convex polygon as their numerical range.

Provides bounds on matrix size for a normal matrix with a specified convex polygon as its higher rank numerical range.

Abstract

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank numerical range $\Lambda_k(A)$ is the intersection of convex polygons with vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1} \le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$ with minimum $n$ such that $\Lambda_k(A) = {\mathcal P}$. In particular, if ${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A \in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $\Lambda_k(A) = {\mathcal P}$.