Does A Have a Positive Eigenvalue Using Topology?

[sin123]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I don't know where to start (besides brute force algebra, maybe).

 

[VeeEight]

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

 

[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!

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