The general solution of the matrix equation wt+∑k=1nwxkρ(k)(w)=ρ(w)+[w,Tρ˜(w)]

We construct the general solution of the equation wt+∑k=1nwxkρ(k)(w)=ρ(w)+[w,Tρ˜(w)], for the N×NN×N matrix w, where T   is any constant diagonal matrix, n,N∈N+n,N∈N+ and ρ(k)ρ(k), ρ  , ρ˜:R→R are arbitrary analytic functions. Such a solution is based on the observation that, as w evolves according to the above equation, the evolution of its spectrum decouples, and it is ruled by the scalar analogue of the above equation. Therefore the eigenvalues of w   and suitably normalized eigenvectors are the N2N2 Riemann invariants. We also obtain, in the case ρ=ρ˜=0, a system of N2N2 non-differential equations characterizing such a general solution. We finally discuss reductions of the above matrix equation to systems of N equations admitting, as Riemann invariants, the eigenvalues of w. The simplest example of such reductions is a particular case of the gas dynamics equations.