Higher-rank Numerical Ranges and Kippenhahn Polynomials

TL;DR

This paper establishes a precise equivalence between the equality of higher-rank numerical ranges of matrices and the coincidence of their Kippenhahn polynomials, using advanced algebraic and spectral tools.

Contribution

It proves that matrices have identical higher-rank numerical ranges for all relevant k if and only if their Kippenhahn polynomials are identical, linking geometric and algebraic matrix invariants.

Findings

Higher-rank numerical ranges determine Kippenhahn polynomials

Kippenhahn polynomials uniquely identify matrices with equal numerical ranges

The proof combines spectral theory, algebraic geometry, and matrix analysis

Abstract

We prove that two n-by-n matrices A and B have their rank-k numerical ranges $\Lambda_k(A)$ and $\Lambda_k(B)$ equal to each other for all k, $1\le k\le \lfloor n/2\rfloor+1$, if and only if their Kippenhahn polynomials $p_A(x,y,z)\equiv\det(x Re A+y Im A+zI_n)$ and $p_B(x,y,z)\equiv\det(x Re B+y Im B+zI_n)$ coincide. The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl's perturbation theorem for eigenvalues of Hermitian matrices and Bezout's theorem for the number of common zeros for two homogeneous polynomials.