Is it true that every infinite product of positive matrices (every element is larger than 0) converges to a rank one matrix?
I learned that Birkhoff's Contraction Coefficient is smaller than one for positive matrices and that Birkhoff's Contraction Coefficient is submultiplicative.
Additionally, Birkhoff's Contraction Coefficient is 0 for rank 1 matrices.
Combining these insights, the claim should follow. But it is somehow against my intuition.
Thanks for your help!