An square matrix is called regular if for some integer all entries of are positive.
is a regular matrix, because has all positive entries.
It can also be shown that all other eigenvalues of A are less than 1, and algebraic multiplicity of 1 is one.
It can be shown that if is a regular matrix then approaches to a matrix whose columns are all equal to a probability vector which is called the steady-state vector of the regular Markov chain.
It can be shown that for any probability vector when
gets large, approaches to the steady-state vector
It can also be shown that the steady-state vector q is the only vector such that
Note that this shows q is an eigenvector of A and is eigenvalue of A.