= Strict hyperbolicity
A first-order system $U_t+A(U)U_x=C(U)$ is strictly hyperbolic at a state when $A$ has distinct real <eigenvalues>. These separate its <characteristic> families and permit independent compatibility equations from <left eigenvectors>. At a repeated <eigenvalue> <strict hyperbolicity> fails; if the <matrix> is also defective, even <hyperbolicity> in the complete-<eigenvector> sense fails. For $A=\bigl(\begin{smallmatrix}u&2\\2v&u\end{smallmatrix}\bigr)$ the <eigenvalues> $u\pm2\sqrt v$ are distinct precisely when $v>0$.
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