Higher-rank numerical ranges and Kippenhahn polynomials

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 <= k <= left perpendicularn/2right perpendicular + 1, if and only if their Kippenhahn polynomials P-A (x, y, z) equivalent to det(xRe A + yIm A + zI(n)) and p(B) (x, y, z) equivalent to det(xRe B + yIm 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. (C) 2012 Elsevier Inc. All rights reserved.