# Homework Help: Non-Negative Matrices

1. Dec 6, 2012

### BrainHurts

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. Dec 6, 2012

### micromass

What do you mean with a "nonnegative vector" or "positive vector"?

3. Dec 6, 2012

### BrainHurts

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
4. Dec 7, 2012

### Ray Vickson

Google Perron-Frobenius theorem.

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