The numerical range of linear pencils of 2-by-2 matrices

AbstractLet A,B∈Mn. The numerical range of a linear pencil Aλ−B is defined byW(Aλ−B)={t∈C:tw*Aw−w*Bw=0forsomenon-zerow∈Cn}.The numerical range of a linear pencil of n-by-n matrices coincides with the union of the numerical ranges of linear pencils of 2-by-2 matrices. We emphasize on computation of the numerical range of linear pencils of 2-by-2 matrices. Let A,B∈M2 with constant diagonals. We show that the boundary of W(Aλ−B) lies on a rational curve of degree at most 4. By using Lagrange's multiplier and implicit function, we describe the boundary of W(Aλ−B) for general 2-by-2 matices