Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in R nWe prove that if λ is a continuous function onDsuch that, for every x ϵD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.