Right. Both the matrix S and the identity matrix will project the stationary distribution onto itself. So any linear combination of them will project the stationary distribution onto itself. Let me know if I'm saying something really stupid
Oh, you meant "multiple of the identity matrix" when you said "diagonal matrix"?
Doesn't that mean you're assuming the number of people staying with any language X is the same regardless of X?
[[1 1] [1 1]] and [[10 1] [1 1]] will have different stationary distributions, the values on the diagonal will likely be different than a multiple of the identity matrix.