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Math Help - Linear system with only positive solutions

  1. #1
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    Linear system with only positive solutions

    Hi,
    I'm involved in the dimensioning of a markov chain.
    In order to find its state probabilities I have constructed a system of equation, resulting in a sparse matrix of coefficients of the type Ax = b.

    The problem is that when I solve the system (the solutions are correct, I already checked it), some states have negative solutions (which means that the probability of being in these states are negative ).

    Is there any rule/condition/property that the matrix A must have in order to obtain only Zero or Positive solutions?

    Thanks in advance,
    G.B.
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  2. #2
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    Quote Originally Posted by pesius View Post
    Hi,
    I'm involved in the dimensioning of a markov chain.
    In order to find its state probabilities I have constructed a system of equation, resulting in a sparse matrix of coefficients of the type Ax = b.

    The problem is that when I solve the system (the solutions are correct, I already checked it), some states have negative solutions (which means that the probability of being in these states are negative ).

    Is there any rule/condition/property that the matrix A must have in order to obtain only Zero or Positive solutions?

    Thanks in advance,
    G.B.
    Can you post what you have done? The probabilities of the columns of a Markov chain matrix are nonnegative and add up to 1.
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  3. #3
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    Quote Originally Posted by dwsmith View Post
    Can you post what you have done? The probabilities of the columns of a Markov chain matrix are nonnegative and add up to 1.
    The probabilities add up to 1, but some of them are negative.

    Reading again my initial post I've seen that the problem is not well explained.

    The linear system is of the type Ax = b, where A is a sparse matrix, b is a column vector of all zeros (only the last element is 1) and x is a column vector that contains the states probabilities.

    The vector x that must contains only positive solutions and I would like to find out some properties that A should respects to guarantee that condition.
    Last edited by pesius; March 28th 2011 at 07:22 AM.
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