Numerical range of linear pencils (Q1587893)

For a linear pencil \(A\lambda +B\) (\(A, B\) are complex \(n\times n\) matrices) the numerical range of \(A\lambda +B\) is defined as \(W(A\lambda +B)=\{\lambda \in {\mathbb{C}}\): \(x^*(A\lambda +B)x=0\) for some nonzero \(x\in {\mathbb{C}}^n\}\), being a generalization of the classical one. The author studies the interplay between the geometrical properties of \(W(A\lambda +B)\) (with emphasis to its boundary) and the algebraic and analytic properties of the pencil \(A\lambda +B\). For instance, it is obtained that the eigenvalues of \(A\lambda +B\) on the boundary of \(W(A\lambda +B)\) are semisimple. Further, the boundary of selfadjoint linear pencils and pencils \(A\lambda +B\) with Hermitian matrix~\(A\) is studied. In the case when either \(A\) or \(B\) are Hermitian an answer to the problem of the numerical approximation of \(W(A\lambda +B)\) is presented. The numerical range of a matrix on an indefinite inner product space is also considered.