# Math Help - Limit of Sums of matrices

1. ## Limit of Sums of matrices

Hello,

I have no idea how to solve this:

$x = (1-\alpha) \sum_{r=0}^\infty \alpha^r W^r Y$

where $x \in \mathbb{R}^n, 0 < \alpha < 1, W \in \mathbb{R}^{n x n}$ symmetric , $Y \in \mathbb{R}^n$.

Derive x in the limit $\alpha \rightarrow 0$.

We are just interested in the sign of $x_i$. It is helpful to expand the sum in $x_i$. What happens as $\alpha \rightarrow 0$? Which terms in the sum are non-zero? Which terms dominate as $\alpha \rightarrow 0$?
Thank You!

2. I am not sure what is intended by the suggestion "to expand the sum in $x_i$."

The series $\sum_{r=0}^\infty \alpha^r W^r$ converges in the operator norm to $(I-\alpha W)^{-1}$ whenever $0<\alpha<\|W\|^{-1}$. Therefore $x = (1-\alpha)(I-\alpha W)^{-1}Y\to Y$ as $\alpha\to0$.

3. Thanks for your answer!

Expanding in $x_i$ means solving this term with repect to the $i$-th component of $x$.

Your post looks plausible to me, but are we allowed to do this? I forgot to mention the following hint in the description:

Consider the limit $\alpha \rightarrow 0$. Setting $\alpha = 0$ will yield a different result!

4. I'm puzzled by that. I'd be interested to see the intended answer, because I don't see how it could be different from the one I suggested. Are you sure that the problem is correctly stated?

5. The problem should be correct. I will write again when I have found something.