  Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral
numbers $\alpha_1,...,\alpha_n\in{\mathbb R}$ to real upper triangular $n\times
n$ matrices $S$ with 1's on the diagonal and eigenvalues of $S^{-1}S^t$ in the
unit sphere. Especially, $\exp(-2\pi i\alpha_j)$ shall be the eigenvalues of
$S^{-1}S^t$.
  We tried to make their idea rigorous, but we succeeded only partially. This
paper fixes our results and our conjectures. For certain subfamilies of
matrices their idea works marvellously, and there the spectral numbers fit well
to natural (split) polarized mixed Hodge structures. We formulate precise
conjectures saying how this should extend to all matrices $S$ as above.
  The idea might become relevant in the context of semiorthogonal
decompositions in derived algebraic geometry. Our main interest are the cases
of Stokes like matrices which are associated to holomorphic functions with
isolated singularities (Landau-Ginzburg models). Also there we formulate
precise conjectures (which overlap with expectations of Cecotti and Vafa). In
the case of the chain type singularities, we have positive results.
  We hope that this paper will be useful for further studies of the idea of
Cecotti and Vafa.
