Geometry of the numerical range of matrices

Some techniques for the study of the algebraic curve C(A) which generates the numerical range W(A) of an n×n matrix A as its convex hull are developed. These enable one to give an explicit point equation of C(A) and a formula for the curvature of C(A) at a boundary point of W(A). Applied to the case of a nonnegative matrix A, a simple relation is found between the curvature of the function Φ(A)=p((1?α)A+ αA^T) (pbeingthePerronroot) at $\text{α=}\text{1}\text{2}$ and the curvature of W(A) at the Perron root of $\text{1}\text{2}\text{(A+A}{}^{\mathrm{T}}\text{)}$. A connection with 2-dimensional pencils of Hermitian matrices is mentioned and a conjecture formulated.