Numerical Range of Linear Pencils

Let A and B be n × 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 − λB, where λ is a complex number. The numerical range of a linear pencil of a pair (A, B) is the set {W(A,B)={x*(A-\lambda B)x : x\in C^{n}, ||x||=1, \lambda\in 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 − λB, when one of the matrices A or B is Hermitian and λ ∈ 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.