Matrix difference equation

Remarks

Example: with $m = 2$, $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ and $b = 0$, the sequence $x(n) = (F_{n+1}, F_n)^T$ of Fibonacci numbers is a solution, since $F_{n+2} = F_{n+1} + F_n$ and the second entry just shifts. Non-example: $x(n+1) = x(n)^2$ for $x(n) \in K$ is not a matrix difference equation, since the right side is not of the form $A(n) x(n) + b(n)$. In the homogeneous constant-coefficient case the solutions are $x(n) = A^n x(0)$. Equations of higher order, such as $x(n+1) = A_0 x(n) + A_1 x(n-1)$, reduce to first order by stacking $(x(n), x(n-1))$ into one vector of size $2m$. In this way a scalar linear recurrence of order $d$ becomes a first-order system with a companion matrix.