Numerical range of linear pencils

AbstractConsider a linear pencil Aλ+B, where A and B are n×n complex matrices. The numerical range of Aλ+B is defined asW(Aλ+B)=λ∈C:x*(Aλ+B)x=0forsomenonzerox∈Cn.In this paper, we study the geometrical properties of W(Aλ+B), with emphasis to its boundary. An answer to the problem of the numerical approximation of W(Aλ+B), when one of the coefficients A and B is Hermitian, is presented. The numerical range of a matrix on an indefinite inner product space is also considered

The polar decomposition of block companion matrices

AbstractLet L(λ) = Inλm + Am−1λm−1 + …+A1λ + A0 be an n × n monic matrix polynomial, and let CL be the corresponding block companion matrix. In this note, we extend a known result on scalar polynomials to obtain a formula for the polar decomposition of CL when the matrices A0 and Σj=1m−1 AjA*j are nonsingular

Numerical approximation of the boundary of numerical range of matrix polynomials

The numerical range of an n×n matrix polynomial P (λ) = Amλm+ Am−1λm−1 + · · ·+A1λ+A0 is defined by W (P) = {λ ∈ C: x∗P (λ)x = 0, x ∈ Cn, x∗x = 1}, and plays an important role in the study of matrix polynomials. In this paper, we describe a methodology for the illustration of its boundary, ∂W (P), using recent theoretical results on numerical ranges and algebraic curves