Hyperbolic forms associated with cyclic weighted shift matrices

The singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined. It is shown that the singular points of such a curve are real nodes. Some results related the numerical ranges of cyclic weighted shift matrices are presented. In particular, the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form. Further, an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights.