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Non-Negative Matrices

  1. Dec 6, 2012 #1
    1. The problem statement, all variables and given/known data
    If A≥0 and Ak>0 for some k≥1, show that A has a positive eigenvector.



    2. Relevant equations



    3. The attempt at a solution
    A is nxn

    Well from a previous problem we know that the spectral radius ρ(A)>0

    We also know that if A≥0, then ρ(A) is an eigenvalue of A and there is a non negative vector x, x=/=0 such that Ax=ρ(A)x

    Kinda stuck
     
  2. jcsd
  3. Dec 6, 2012 #2

    micromass

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    What do you mean with a "nonnegative vector" or "positive vector"?
     
  4. Dec 6, 2012 #3
    non negative vector means all the entries in that vector is greater than zero,

    if the vector is positive all entries in that vector is positive

    i.e. if x≥0 all components of x are greater than or equal to zero

    similarly if a matrix A≥0

    all [aij]≥0

    positive just means everything is greater than 0
     
    Last edited: Dec 6, 2012
  5. Dec 7, 2012 #4

    Ray Vickson

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    Google Perron-Frobenius theorem.
     
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