Numerical Range of Linear Pencils

Let A, B be n X n (complex) matrices. We are mainly interested in the study of the structure of the spectrum of a linear pencil, that is, a pencil of the form A – B, where is a complex number. The numerical range of a linear pencil of a pair (A, B) is the set W(A, B) = {x * (A - B) x:x Cn, x 1, C}. The numerical range of linear pencils with Hermitian coefficients was studied by some authors. We are mainly interested in the study of the numerical range of a linear pencil, A - B, when one of the matrices A or B is Hermitian and C. we characterize it for small dimensions in terms of certain algebraic curves. For the case n = 2, the boundary generating curves are conics. For the case n = 3, all the possible boundary generating curves can be completely described by using Newton’s classification of cubic curves. The results are illustrated by numerical examples.