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We show that if A is an n-by-n (n⩾3) matrix of the form0a10⋱⋱an-1an0,then the boundary of its numerical range contains a line segment if and only if the aj’s are nonzero and the numerical ranges of the (n-1)-by-(n-1) principal submatrices of A are all equal. For n=3, this is the case if and only if |a1|=|a2|=|a3|≠0, in which case W(A), the numerical range of A, is the equilateral triangular region...
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