# Stability of $x(n) = A x(n-1)+b$

I am looking at the following system: $$x(n) = A x(n-1) + b$$ where x and b are vectors and A is a matrix. How can I derive the stability and causality conditions for such a system using Z transform?

If I apply Z transform to each element of x, I have: $$X(z) = A z^{-1} X(z) + b$$ Therefore, $$(I - Az^{-1})X(z) = b$$. Then, $$X(z) = (I - Az^{-1})^{-1} b$$.

Is this correct? What are the conditions for stability and causality?

• Is $b$ time-varying, or did you perhaps mean $x_n = A x_{n-1} + b u_n$, where $u_n$ is the system input? – TimWescott Sep 30 '20 at 17:29
• This looks so very much like the canonical state-space representation (aside from $b$ not being a constant multiplier to an input $u_n$) that I'm wondering -- is it homework? – TimWescott Sep 30 '20 at 17:30