# Limit of two matrices each to the kth power and multiplied

1. Jul 4, 2013

### JPBenowitz

If I have two square nonnegative primitive matrices where the Perron-Frobenius Theorem applies how would I calculate lim (A^k)(B^k) as k approaches infinity.

2. Jul 4, 2013

### JPBenowitz

I'm assuming I would just find A^k and B^k by diagonalization multiply them and then find the limit?

3. Jul 4, 2013

### Simon Bridge

You would probably be expected to exploit the properties of those non-negative primitive matrices which P-F can be applied to. I'd start by working though the definitions - bearing in mind what you understand about matrix manipulations. Note: there is a fair amount of overlap between the three conditions:

i.e.
is it possible to have a primitive matrix that is not non-negative?
what sorts of matrices can F-P be applied to?