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We study bi-Hamiltonian systems of hydrodynamic type with non-singular(semisimple) non-local bi-Hamiltonian structures and prove that such systems ofhydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrarynon-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamictype, there exist local coordinates (Riemann invariants) such that all therelated matrix differential-geometric objects, namely, the matrix V^i_j(u) ofthis system of hydrodynamic type, the metrics g^{ij}_1(u) and g^{ij}_2(u) andthe affinors (w_{1, n})^i_j(u) and (w_{2,n})^i_j(u) of the non-singularnon-local bi-Hamiltonian structure of this system, are diagonal in these localcoordinates. The proof is a natural consequence of the general results of thetheory of compatible metrics and the theory of non-local bi-Hamiltonianstructures developed earlier by the present author.

On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type