How Does a Non-Negative Matrix Ensure a Positive Eigenvector?

[BrainHurts]

Homework Statement

If A≥0 and A^k>0 for some k≥1, show that A has a positive eigenvector.

Homework Equations

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

 

[micromass]

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

 

[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 [a_ij]≥0

positive just means everything is greater than 0

 

[Ray Vickson]

Homework Statement

If A≥0 and A^k>0 for some k≥1, show that A has a positive eigenvector.

Homework Equations

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

Google Perron-Frobenius theorem.