| Résume | In this work, we study the regularities of eigenvalues and eigenvectors of k-parameter linear families of real symmetric matrices A(t), t ∈ R^k, following, among others, works of Rellich (1937) and Kurdyka-Paunescu (2008).
We introduce the monodromy of A(t), which is the action of the first homotopy group of regular parameters on the spectrum of A(t), and the related antipodal monodromy. We first characterize all possible monodromies, and in particular, realize any permutation as the antipodal monodromy of a 2-family. We then study the analytic reductions of A(t) , and prove
that the existence of a non trivial antipodal monodromy is the only obstruction to get full diagonalization of A(t) . Finally, we study the couples eigenvalues/eigenvectors
for 2-families in Sym_3(R), and prove that those families are classified by the given of one couple of type of cubic curves of P^2 among nine possibilities. |
