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Math Help - Probability Vectors

  1. #1
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    Probability Vectors

    Show that when x is a probability vector and A is a stochastic matrix, then Ax is another probability matrix
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  2. #2
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    Re: Probability Vectors

    You mean "then Ax is another probability vector".

    What are your precise definitions of "probability vector" and "stochastic matrix"?
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  3. #3
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    Re: Probability Vectors

    Yes, sorry.

    We say a vector X in Rn is a probability vector if all its entries are non-negative and add up to 1. We say a square matrix is a stochastic matrix if each of its column vectors is a probability vector
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  4. #4
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    Re: Probability Vectors

    clearly each coordinate of Ax is non-negative being the sum of products of non-negative numbers.

    let us call the i,j-th entry of A, aij.

    then (Ax)i = ai1x1 +...+ ainxn.

    hence (Ax)1 +...(Ax)i +...+ (Ax)n =

    (a11x1 +...+ a1nxn) +...+ (ai1x1 +...+ ainxn) +...+ (an1x1+ ...+ annxn) =

    (a11 +...+ an1)x1 +...+ (a1j +...+ anj)xj +...+ (an1 +...+ ann)xn.

    but a1j +...+ anj = 1, since A is a stochastic matrix (this is just the sum of the j-th column, which is a probability vector), for EACH j,

    hence (Ax)1 +...+ (Ax)n = (a11 +...+ an1)x1 +....+ (an1 +...+ ann)xn = x1 +... + xn = 1, so Ax is a probability vector.
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