I am looking for an analytical solution for the linear PDE

$(1)\qquad\qquad \qquad f\_t+ A f\_x + B f = 0, $

Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector. 

Clearly each one of $f\_t + A f\_x = 0,$ and $f\_t+ B f = 0,$ has an analytical solution involving eigenvectors and eigenvalues of $A$ and $B$ separately, but I'm not sure about equation (1).

My question is:

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> What are the minimal conditions on $A$ and $B$ such that there is an analytical solution to the problem in (1), (clearly if they share eigenvectors, I can write a solution...) and what is the solution in that case?
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