Second order sensitivity analysis of all eigenvalues of a symmetric matrix

Given a symmetric matrix A(x) = [a(ij)(x)] depending on a parameter x, we study the second order sensitivity of all the eigenvalues lambda(m)(x) of A(x). Under some smoothness assumptions like the a(ij) being twice differentiable, we prove that the second order directional derivatives (which naturally arise in the context of nonsmooth functions) of lambda(m) do exist, are equal, and we give their explicit expression in terms of the data of the parametrized matrix. The techniques used are quite different from those for the first order sensitivity; they rely on sharp results by G.W. Stewart concerning the perturbation of invariant subspaces.