how to measure if a matrix reach it limit in an iteration process?

Question

There is a iteration process : S_k+1 =BS_kA^T+B^TS_kA

S is the similarity matrix of G_A and G_B. And A and B are adjacency matrices of graphs and S₀ is a matrix whose entries are all equal to 1. Sequence S_k does not always converge, but the even and odd subsequences do converge.

I want to know how to measure if S_k reach it limit in the iteration process?