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Math Help - Limit of Sums of matrices

  1. #1
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    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!
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  2. #2
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    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.
    Last edited by Opalg; January 26th 2009 at 11:51 AM.
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  3. #3
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    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!
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  4. #4
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    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?
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  5. #5
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    The problem should be correct. I will write again when I have found something.
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