Trace conditions for symmetry of the numerical range

Abstract

Abstract. A subset S of the complex plane has n-fold symmetry about the origin (n-sato) if z ∈ S implies e 2π n z ∈ S. The 3 × 3 matrices A for which the numerical range W (A) has 3-sato have been characterized in two ways. First, W (A) has 3-sato if and only if the spectrum of A has 3-sato while tr(A 2 A * ) = 0. In addition, W (A) has 3-sato if and only if A is unitarily similar to an element of a certain family of generalized permutation matrices. Here it is shown that for an n × n matrix A, if a specific finite collection of traces of words in A and A * are all zero, then W (A) has n-sato. Further, this condition is shown to be necessary when n = 4. Meanwhile, an example is provided to show that the condition of being unitarily similar to a generalized permutation matrix does not extend in an obvious way.