TY - JOUR

T1 - Higher-rank numerical ranges and Kippenhahn polynomials

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (P.Y. Wu). 1 The research of Hwa-Long Gau is supported in part by the National Science Council of the Republic of China under project NSC 101-2115-M-008-006. 2 The research of Pei Yuan Wu is supported in part by the National Science Council of the Republic of China under project NSC 101-2115-M-009-004 and by the MOE-ATU.

PY - 2013/4/1

Y1 - 2013/4/1

N2 - We prove that two n-by-n matrices A and B have their rank-k numerical ranges Λk(A) and Λk(B) equal to each other for all k,1≤k≤⌊n/2⌋+1, if and only if their Kippenhahn polynomials pA(x,y,z)≡det(xReA+yImA+zIn) and pB(x,y,z)≡det(xReB+yImB+zIn) 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 Bézout's theorem for the number of common zeros for two homogeneous polynomials.

AB - We prove that two n-by-n matrices A and B have their rank-k numerical ranges Λk(A) and Λk(B) equal to each other for all k,1≤k≤⌊n/2⌋+1, if and only if their Kippenhahn polynomials pA(x,y,z)≡det(xReA+yImA+zIn) and pB(x,y,z)≡det(xReB+yImB+zIn) 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 Bézout's theorem for the number of common zeros for two homogeneous polynomials.

KW - Higher-rank numerical range

KW - Kippenhahn polynomial

UR - http://www.scopus.com/inward/record.url?scp=84873707835&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.11.017

DO - 10.1016/j.laa.2012.11.017

M3 - 期刊論文

AN - SCOPUS:84873707835

SN - 0024-3795

VL - 438

SP - 3054

EP - 3061

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 7

ER -