I was given the following as an example of a linear recurrence and I don't understand how it works...
Let us call the following eq1: $x_i = \begin{bmatrix} \sum_{z = 1}^i{zk^{z-1}} \\ (i+1)k^i \\ k^i \end{bmatrix}$
And the following eq2:
$ x_{i+1} = \begin{bmatrix} 1 & 1 & 0 \\ 0 & k & k \\ 0 & 0 & k \end{bmatrix}x_i $
Given eq1, by what steps, method or algorithm can you derive eq2?
Likewise for the reverse, by what method can you derive eq1 from eq2?