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Math Help - Eigen space and probability question

  1. #1
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    Eigen space and probability question

    Hi. I have a bit of a hard time understanding how to answer the following question:

    Given to cities A and B located close to one another. There is a certain probability that someone will move from A to B during one year, and a certain probability that someone will move from B to A during the same year. Is there an eigen space independent of the probabilities?

    Any help would be greaty appreciated . I'm quite stuck on how to answer this.
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  2. #2
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    Quote Originally Posted by krje1980 View Post
    Hi. I have a bit of a hard time understanding how to answer the following question:

    Given to cities A and B located close to one another. There is a certain probability that someone will move from A to B during one year, and a certain probability that someone will move from B to A during the same year. Is there an eigen space independent of the probabilities?

    Any help would be greaty appreciated . I'm quite stuck on how to answer this.
    Here is a hint:

    Let p be the probability that someone In A moves from A to B and
    let q be the probability that someone in B moves from B to A then

    you get the matrix

    \begin{bmatrix} (1-p) & p \\ q & (1-q) \end{bmatrix}

    Now try to find the eigenvalues and eigenvectors and see what happens
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  3. #3
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    OK.

    So just for fun I chose p = 0,2 and q = 0,3. I get eigenvalues 1 and 0,5. Eigenvectors are (1, 1) and (-1, 3/2).

    But how does this help me answer the question? Is it because I get 1 as an eigenvalue, there can not exist an eigen space independent of the probabilities?
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  4. #4
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    Anybody who knows? I really appreciate any help
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  5. #5
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    Well, if you are doing this "just for fun", I don't see why anyone would bother to respond. If it is serious, then it probably would have been better to do what TheEmptySet suggested: actually find the eigenvectors of \left|\begin{array}{cc}1- p & p \\ q & 1- q\end{array}\right|, rather that picking simple values for p and q.

    If you had done that, you would learned that the eigenvalues of any such matrix are 1 and 1- (p+q). In particular, except in the trivial case p= q= 0, those are distinct and so they have two independent eigenvectors- the eigenspace is all R^2.

    If p= q= 0, then the matrix is \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} which has the single eigenvalue 1 but since \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x \\ y \end{bmatrix}, it is still true that every vector is an eigenvector and the eigenspace is, again, all of R^2.
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  6. #6
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    Dude, I'm not doing the problem "for fun." I'm sorry if my choice of words made it seem like that. As I mentioned in the first post, I was unsure how to complete the problem. Based on the suggestion given to me I picked two values, one for p and one for q, in order to see if that could give me any sense of what the answer could be. After all, I'm still finding eigen values for the system. I just started learning about eigen values one week ago, so I'm sorry if I didn't "get" exactly what I was supposed to do.

    Nevertheless, I really appreciate your answer. Thanks a lot
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