Numerical ranges of companion matrices Original Research Article

We show that an n-by-n companion matrix A can have at most n line segments on the boundary ∂W(A) of its numerical range W(A), and it has exactly n line segments on ∂W(A) if and only if, for n odd, A is unitary, and, for n even, A is unitarily equivalent to the direct sum A1 circled plus A2 of two (n/2)-by-(n/2) companion matricesimage with 1 less-than-or-equals, slant midamid < tan(π/n) + sec(π/n).