Suppose the output $y_h[n]$ of a linear, time-invariant system is described by by the following equation for input $x[n]=0$,
$$\sum_{k=0}^{N}a_k y_h[n-k] = 0$$
My book states the $y_h[n]$ is in fact a member of a family of solution of the form:
$$y_h[n]=\sum_{m=1}^{N}A_mz_m^n$$
where the complex numbers $z$ are the zeros of $A(z)=\sum_{k=0}^Na_kz^{-k}$.
What is the proof that $y_h$ is in fact of the form given?
A simple substitution would result in:
$$\sum_{k=0}^N (a_k \sum_{m=1}^NA_mz_m^n) = 0$$
but I don't see to proof showing a set $\{z_1,z_2,...,z_N\}$ and $\{A_1, A_2,..., A_N\}$ indeed exist for all n. Sorry if this seems like a poor question.