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Eigenvalue via topology?

  1. Feb 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A denote a 3x3 matrix with positive real entries. Show that A has a positive real Eigenvalue.


    2. Relevant equations

    This is a problem from a topology course, assigned in the chapter on fundamental groups and the Brouwer fixed point theorem.


    3. The attempt at a solution

    I don't know where to start (besides brute force algebra, maybe).
     
  2. jcsd
  3. Feb 3, 2010 #2
    I believe that you do in fact need to use Brouwer's Fixed Point Theorem here.
     
  4. Feb 3, 2010 #3
    I figured that much, if just for purely pedagogical reasons.

    For a while I didn't know how to use the positive entries of the matrix, until I realized that that means that the first octant is mapped to itself by the linear transformation. Follow the linear transformation by a projection and I am set up for Brouwer.

    Done!

    <('-')> <(''<) <('-')> (>'-')> <('-')> <(''<) <('-')> (>'-')> <('-')> <(''<) <('-')> <('-')> (>'-')> <('-')> <(''<) <('-')>
     
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