I want to show that the shallow water equations of the form $$\partial_t\left(\begin{array}{c} h\\ v \end{array}\right)+\partial_x\left(\begin{array}{c} hv\\ \frac{1}{2}v^2+gh \end{array}\right)=0 $$ are hyperbolic, where $h>0$ and $g$ is the acceleration of gravity.
So I have to show that the Jacobi of $f(\left( \begin{array}{c} h\\ v\\ \end{array} \right))$ is diagonalizable. So I computed $$df=\left( \begin{array}{c} v~~~~h\\ g~~~v\\ \end{array} \right)$$ but it is not diagonalizable since the characteristic polynomial does not factor into linear factors. Can anyone help me?