Subdifferentials (limiting Fréchet, Clarke) of the composition \(f\circ \lambda \) of extended real valued permutation invariant functions \(f\) and the eigenvalue vector function \(\lambda \) of a symmetric matrix \(X\) are calculated using the transformation to principles axes. If \(U\) is an orthogonal matrix, such that \(\text{Diag} ( \lambda ( X)) =UXU^{T}\) is the diagonal transform of \(X\) and \(f( Px) =f( x) \) for each permutation matrix \(P\) then under assumptions usual in nonsmooth analysis for \(f\) it is shown \(\partial ( f\circ \lambda) ( X) =\{ U^{T} \text{Diag}( \mu) U\mid U\) as above, \(\mu \in \partial f( x) |_{x=\lambda ( X)}\}\). In this paper the motivation is underlined and the principal strategy is developed without proofs. The full paper with all proofs can be found in Math. Program. 84 A, No. 1, 1-24 (1999; following review).