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, $1\le k\le \lfloor n/2\rfloor +1$, if and only if their Kippenhahn polynomials $p_A(x,y,z)\equiv \det (xReA+yImA+z{I}_n)$ and $p_B(x,y,z)\equiv \det (xReB+yImB+z{I}_n)$ 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.

16 pages, 1 figure

References in corpus (3)

• Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
• Condition for the higher rank numerical range to be non-empty
• Higher rank numerical ranges of normal matrices

Cited by in corpus (2)

• Invariant hyperbolic curves: determinantal representations and applications to the numerical range
• On the ellipticity of the higher rank numerical range