A remark on the generalized numerical range of a normal matrix

Abstract

Let A be an n × n complex normal matrix and let (A) = {diag UAU*: U is unitary) where U* is the conjugate transpose of U. It is known that (A) may not be convex [1, 3] and it is convex when A is Hermitian [1, 2]. In this note we show that (A) is convex if and only if the eigenvalues of A are collinear (i.e. there exist complex numbers α ( ≠ 0) and β such that αA + βi is Hermitian).