Higher-rank Numerical Ranges and Kippenhahn Polynomials

We prove that two n-by-n matrices A and B have their rank-k numerical ranges ${\mathrm{\Lambda }}_k(A)$ and ${\mathrm{\Lambda }}_k(B)$ equal to each other for all k, $1\le k\le \lfloor n/2\rfloor+1$,ifandonlyiftheirKippenhahnpolynomials$p_A(x,y,z)\equiv\det(x Re A+y Im A+zI_n)$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.