The numerical range of self-adjoint quadratic matrix polynomials (Q2784369)

Let \(P(\lambda)=A_2\lambda^2+A_1\lambda+A_0\) be an \(n\times n\) self-adjoint quadratic matrix polynomial, where \(A_0,A_1,A_2\) are \(n\times n\) Hermitian matrices and \(\lambda\in {\mathbb C}\). Let \(W(P)=\{\mu\in {\mathbb C}\mid x^*P(\mu)x=0\) for some nonzero \(x\in {\mathbb C}^n\}\), and let \(F(A)\) be the (classical) numerical range of~\(A\). The paper presents a discussion of properties of~\(W(P)\) with emphasis on the connection with the spectrum of \(P(\lambda)\), and also on monic polynomials. A~classical result stating that non-smooth points of the boundary~\(\partial F(A)\) are necessary eigenvalues of~\(A\) is generalized onto \(\partial W(P\)). Some extensions of the theory to more general polynomials \(P(\lambda)\) are also discussed, as well as special cases describing vibrating systems.