On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial Original Research Article

We describe a connection between the geometrical structure of the numerical range W(L) of a selfadjoint quadratic operator polynomial L(λ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m,n (1less-than-or-equals, slantmless-than-or-equals, slant2n) we construct a monic quadratic polynomial Lm,n(λ)=λ2+λC+B with selfadjoint n×n matrices C, B which has an eigenvalue λ0set membership, variant∂W(Lm,n) with a Jordan chain of length m.