# Eigenvalue via topology?

1. Feb 3, 2010

### sin123

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. Feb 3, 2010

### VeeEight

I believe that you do in fact need to use Brouwer's Fixed Point Theorem here.

3. Feb 3, 2010

### sin123

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!

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