It's well-known how to find the general solution to a linear recurrence $$ a(t) = c_1a(t-1) + \cdots + c_ka(t-k) $$ by looking at the characteristic polynomial, etc.
What is the general method to solve a linear recurrence relation involving vectors and matrices as coefficients? Here $v(t) \in \mathbb C^{n}$ and $A_i\in \text{Mat}^{n\times n}(\mathbb C)$.
$$v(t) = A_1v(t-1) + \cdots + A_{t-k}v(t-k).$$