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Math Help - solution to infinite dimensional matrix

  1. #1
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    solution to infinite dimensional matrix

    Consider the equations Ax=0 where both the number of columns and rows of A are countably infinite and all entries are either 1, 0 or -1.

    Is the following statement true or false?

    Ax=0 has a nonnegative bounded solution (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity)

    iff

    Ax=0 has a nonnegative bounded solution with at most finitely many nonzeros (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity AND xi=0 for all but finitely many i).

    I guess the answer is no.

    I greatly appreciate any reference. I have checked a few books but can't find the answer.
    Last edited by vivian6606; July 16th 2011 at 09:02 AM.
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  2. #2
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    Re: solution to infinite dimensional matrix

    Quote Originally Posted by vivian6606 View Post
    Consider the equations Ax=0 where both the number of columns and rows of A are countably infinite and all entries are either 1, 0 or -1.

    Is the following statement true or false?

    Ax=0 has a nonnegative bounded solution (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity)

    iff

    Ax=0 has a nonnegative bounded solution with at most finitely many nonzeros (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity AND xi=0 for all but finitely many i).

    I guess the answer is no.

    I greatly appreciate any reference. I have checked a few books but can't find the answer.
    Suppose that A looks like this:

    A = \begin{bmatrix}1&-1&-1&-1&\ldots\\ 0&1&-1&-1&\ldots\\ 0&0&1&-1&\ldots\\ \vdots&\vdots&&\ddots&\ddots\end{bmatrix},

    with 1s on the main diagonal, 1s everywhere above it and 0s everywhere below it. Then Ax=0, where x = \bigl(\tfrac12,\tfrac14,\tfrac18,\ldots,2^{-n},\ldots\bigr).
    But if Ty = 0 with y_n=0 whenever n≥N, then by looking at the (N1)th coordinate of Ay you can see that y_{N-1}=0, and by "backwards induction" y_n=0 for all n.
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  3. #3
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    Re: solution to infinite dimensional matrix

    THANK YOU SO MUCH!!!

    That helps a lot.
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