On matrices whose numerical ranges have circular or weak circular symmetry

Abstract

In [18] among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W(B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m⩾2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W(A) is a circular disk centered at the origin, then so is W(B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W(A), then λ is a sharp point of W(A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic. Open questions are posed at the end.