Definition
A system of first order linear homogeneous PDEs can be written in the general matrix form as
$$ A_0 \mathbf u_t + A_1 \mathbf u_x + A_2 \mathbf u_y + A_3 \mathbf u_z = 0, $$where $x, y, z, t$ are independent variables, $\mathbf u = (u_1, \ldots, u_N)^T$ is an $N$-dimensional vector of unknown functions, and $A_i$ are $M \times N$ matrices whose entries depend on the independent variables $x,y,z,t$ together with the components of $\mathbf u$. Both $M$ and $N$ are positive integers satisfying $M \geq N$.
$\blacksquare$
I think that they are called of hydrodynamic type or dispersionless when the matrices depends only on the components of $\mathbf u$ (see, for instance, @sergyeyev2018new).
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Author of the notes: Antonio J. Pan-Collantes
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