A range associated with skew symmetric matrix

Abstract

We study the range

where A is an n × n complex skew symmetric matrix. It is a compact convex set. Power inequality s(A^2k+1) ≤ s^2k+1(A), k ∈ ℕ, for the radius s(A) ≔ max_ξ∈S(A) |ξ| is proved. When n = 3, 4, 5, 6, relations between S(A) and the classical numerical range and the k-numerical range are given. Axiomatic characterization of S(A) is given. Sharp points and extreme points are studied.

Keywords:

• numerical range
• skew symmetric matrix
• convexity
• extreme point
• sharp point

AMS Subject Classification:

• Primary 15A60