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There are two common definitions of hyperbolic PDEs:

The first one is commonly found in introductory courses on PDEs and states that second-order (quasi-)linear PDEs are hyperbolic if, for an equation given in the form $$L u = \sum_{i=1}^n \sum_{j=1}^n a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} + \text{lower order terms} = 0,$$ the coefficient matrix $(a_{ij})_{ij}$ has one negative eigenvalue and all others are positive (or the other way around).

The other is common for introductions to numerical methods for first-order hyperbolic systems, where for an equation of the form (with only 1 space variable for the sake of simplicity) $$\partial_t q + A(q) \partial_x q = 0,$$ the problem is said to be hyperbolic if $A$ has only real eigenvalues.

The practical similarities of the two are clear to me: Equations of both types describe the evolution of waves propagating with finite velocities. But I wonder if and how these two definitions are formally related. (In particular, I wonder if the first definition can somehow be applied to the Euler equations of fluid mechanics?)

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    $\begingroup$ See my answer here, or this EoM entry. For your final question about fluids: yes, it is well-known in the irrotational case with no vacuum Euler equation reduces to a quasilinear wave equation; one modern reference is Christodoulou-Miao, Chapter 1. In the case with vorticity the vorticity is transported by the fluid velocity and so hyperbolicity is more delicate, and certainly cannot be treated by your "first definition". $\endgroup$
    – Willie Wong
    Commented Sep 19, 2016 at 15:31

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