Point equation of the boundary of the numerical range of a matrix polynomial Original Research Article

The numerical range of an n×n matrix polynomial image P(λ)=Amλm+Am−1λm−1+cdots, three dots, centered+A1λ+A0is defined by image For the linear pencil P(λ)=Iλ−A, the range W(P) coincides with the numerical range of matrix A, image . In this paper, we obtain necessary conditions for the origin to be a boundary point of F(A). As a consequence, an algebraic curve of degree at most 2n(n−1)m, which contains the boundary of W(P), is constructed.