The Numerical Range of Self-Adjoint Quadratic Matrix Polynomials

Abstract

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.