Recursive relation with matrices

Question

Let $B\in\mathbb{M}_n(\mathbb{R})$ and $b\in\mathbb{R}^n$, then consider the recursive relation: $$x_{n+1}=Bx_n+b$$ I want to determine the conditions for which this scheme convereges for any initial vector $x_0$. I can see that $$x_{n+1}=B^nx_0+(I+B+B^2+\cdots+B^{n})b$$ Now $(I+B+B^2+\cdots+B^{n})b\rightarrow (I-B)^{-1}b$, if $I-B$ is invertible. If the limit is $x$ then $\left[\lim_{n\rightarrow\infty}(I-B^n)\right]x=(I-B)^{-1}b$. What then?

Answer 1

You just need $\rho(B)$ (the spectral radius of $B$) to be strictly less than one. In this case, (i) $B^n x_0 \rightarrow 0$ as $n \rightarrow \infty$ and (ii) $I - B$ is invertible since $\lambda = 1$ is not an eigenvalue of $B$.