Why does Osher use eigenvector decomposition to integrate instead of using the vectors itself for integrating?

i'm writing directly from the following paper, pages 68-69

"J.Casper, H.L.Atkins, A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems, Journal of Computational Physics, 106, 62-76"

"(...) It would therefore seem natural, purely from the viewpoint of approximation theory, to reconstruct the set {u_ij} by applying the scalar reconstruction R^2 in a component-wise fashion. However, this approach is valid only if we assume that u(x,y,t) remains free of discontinuities for the duration of a desired calculation. But in the more general case, when u(x,y,t) is a solution of a non-linearly coupled system of equations, such a solution can admit the collision of discontinuites of the same or of different families, as well as their collision with boundaries, i.e. solid walls. In the vicinity of such collisions, a component-wise reconstruction may develop spurious oscillations during this brief encounter which do not dissipate as the discontinuites then distance themselves from one another, if indeed they ever do. (...) It is argued in [6] that, in the constant-coefficient case, it is adventageous to use these characteristic variables in the reconstruction procedure, rather than u itself. This is do to the fact that, under the transformation, the coupled system becomes an independent set of equations, thereby rendering any discontinuity in a particular characteristic variable -undetectable- by another."

hope this helps