Also the matrix is negative definite on account on the concavity (4) of with respect to . This is so, because the defining equation of , viz.
A quasilinear system of the type (23) with symmetric coefficient matrices, of which the temporal one is definite, is called symmetric hyperbolic. We conclude that symmetric hyperbolicity of the equations (23) for the fields is equivalent to the concavity of the entropy density in terms of the fields of densities .
Hyperbolicity implies finite characteristic speeds, and symmetric hyperbolic systems guarantee the well-posedness of initial value problems, i.e. existence and uniqueness of solutions – at least in the neighbourhood of an event – and continuous dependence on the data.
Thus without having actually calculated a single characteristic speed, we have resolved Cattaneo’s paradox of infinite speeds. The structure of extended thermodynamics guarantees that all speeds are finite; no paradox can occur!
The fact that a system of balance-type field equations is symmetric hyperbolic, if it is compatible with the entropy inequality and the concavity of the entropy density was discovered by Godunov  in the special case of Eulerian fluids. In general this was proved by Boillat . Ruggeri & Strumia  have found that the symmetry is revealed only when the Lagrange multipliers are chosen as variables; these authors were strongly motivated by Liu’s results of 1972 and by a paper by Friedrichs & Lax  which appeared a year earlier.
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