According to Wikipedia, the discrete-time dynamical system $$x_{k+1} = A\,x_{k} + b$$ with $A$ a matrix and $b$ a vector may be changed to one without the $b$ term via a change of coordinates $$x \to x + (1 - A)^{-1}\,b.$$
(I henceforth assume that the $1$ is supposed to be an identity matrix $I$.) How exactly does one perform this change of coordinates? My initial thought was to introduce a coordinate $y$ and set $$x = y + (I - A)^{-1}\,b$$ and substitute it into the above iteration equation. That gives $$x_{n+1} = A\,y_{n} + A\,(I - A)^{-1}\,b + b$$ which doesn't make a lot of sense to me.
Just figured it out!
Wherever you see an $x$, replace it with $x + (I - A)^{-1}\,b$. In particular, for the evolution equation you obtain \begin{align*} x_{k+1} + (I - A)^{-1}\,b &= A\,(x_{k} + (I - A)^{-1}\,b) + b \\ x_{k+1} &= A\,x_{k} + (A - I)(I - A)^{-1}\,b + b \\ &= A\,x_k - b + b \\ &= A\,x_k. \end{align*}
You're welcome, past me.