Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in series or something like that, also no friction). I was told that if I have $n$ oscillators in my coupled system, then I will always have $n$ distinct normal modes corresponding to this system.
For that to be the case, though, my coefficient matrix would need to have $n$ distinct eigenvalues which correspond to each normal mode. Going further, if my system had $n$ distinct normal modes it would mean that all of my normal modes could be decoupled since my coefficient matrix could be diagonalized by a suitable change of basis, thus suggesting that all linear coupled systems with constant coefficients have normal modes which are independent from one another.
My question pertains to how we could say in general that our matrices will have $n$ distinct eigenvalues as from what I am aware, generally speaking that may not be the case. Any help would be appreciated in understanding this generalization.