The numerical range of selfadjoint matrix polynomials
Peter Lancaster, John Maroulas, Peter Zizler · Birkhäuser Basel eBooks · 1998
The numerical range of a selfadjoint matrix polynomial \( L\left( \lambda \right) = \sum olimits_j^\ell { = 0{\lambda ^j}{A_j}} \) is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.