On eigenvalues of perturbed quadratic matrix polynomials

Abstract

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det(λ 0 2 Iλ0 C+K)=0 for some λ0≠0 (resp. λ0=0), then the Newton diagram of the polynomialt(λ, e) = det(λ2 I+λ(1+e)C+K expanded in (λ−λ0) and e, has at least a point on or below the linex+y=b (resp. has no expanded in (λ−λ0) and e, has at least a point on or below the of 0 as an eigenvalue of λ 0 2 I+λ0 C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veselic proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler