Differentiable eigenvalues of perturbed quadratic matrix functions

Let T(lambda,epsilon) = lambda(2) + lambda C + lambda epsilon D + K be a pe rturbed quadratic matrix polynomial, where C, D, and K are n x n hermitian matrices. Let lambda(0) be an eigenvalue of the unperturbed matrix polynomi al T(lambda,0). With the falling part of the Newton diagram of det T(lambda , epsilon), we find the number of differentiable eigenvalues. Some results are extended to the general case L(lambda, epsilon) = lambda(2) + lambda D( epsilon) + K, where D(epsilon) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(lambda(0),0), then every eige nvalue lambda(epsilon) Of L(lambda,epsilon) near lambda(0) is analytic.