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 $A=\left(\begin{array}{cc}u& 2\\ 2v& u\end{array}\right)$ the eigenvalues $u\pm 2\sqrt{v}$ are distinct precisely when $v>0$.