  We revisit the theory of first-order quasilinear systems with diagonalizable
principal part and only real eigenvalues, what is commonly referred to as
strongly hyperbolic systems. We provide a self-contained and simple proof of
local well-posedness, in the Hadamard sense, of the Cauchy problem. Our
regularity assumptions are very minimal. As an application, we apply our
results to systems of ideal and viscous relativistic fluids, where the theory
of strongly hyperbolic equations has been systematically used to study several
systems of physical interest.
