On flat portions on the boundary of the numerical range Original Research Article

The paper is devoted to matrices with flat portions on the boundary of their numerical range. A constructive criterion for such portions to exist is obtained in case of tridiagonal matrices, and a particular case of continuant matrices is considered. As an application, the cases of (arbitrary) 3 × 3 and 4 × 4 matrices are treated. It is shown, in particular, that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 × 4 matrices is four (three, if the matrices are assumed unitarily irreducible).