I am learning Numerical PDE and came across this part in it.
**Part in the material **
A system of n conservation law is called hyperbolic, if the following is fulfilled: Let $$ A_{i}(u):=D F_{i}(u)=\left(\frac{\partial F_{i}^{(j)}(u)}{\partial u_{k}}\right)_{1 \leq j, k \leq n} $$ If for any $u \in \mathbb{R}^{n}$ and all $\omega \in \mathbb{R}^{d}, \omega \neq 0$ the matrix $$ A(u, \omega)=\sum_{i=1}^{d} \omega_{i} A_{i}(u) $$ has $n$ real eigenvalues $\lambda_{1}(u, \omega) \leq \cdots \leq \lambda_{n}(u, \omega)$ and $n$ linear independent corresponding eigenvectors $v_{1}, \ldots, v_{n},$ i.e. $$ A(u, \omega) v_{k}=\lambda_{k} v_{k}, \quad 1 \leq k \leq n $$ then the system is called hyperbolic. If all eigenvalues are different, then the system is called strictly hyperbolic.
Doubt 1)What does it mean to have n linearly independent eigenvalues in a system, can someone explain this geometrically as well
2)Why does it become strictly hyperbolic when eigenvalues are different? is it due to the fact that the eigenvectors automatically become independent in that case?
3)Can someone tell me the whole concept in simple words or geometrically?