Theorems from Markov systems

Every transition matrix has $1$ as an eigenvalue.

Suppose that a transition matrix $A$ satisfies:

1. $1$ is an eigenvalue with algebraic multiplicity $1$;
2. every other eigenvalue $\lambda_j$ has $|\lambda_j|<1$;
3. there is a basis of eigenvectors for $\mathbb R^n$.

Then $\mathbf{x}(k+1)=A\mathbf{x}(k)$ has a limit $\mathbf{p}$ in the eigenspace for $\lambda=1$. It is the unique vector in that eigenspace whose component sum equals the component sum of $\mathbf{x}(0)$; it may be zero when that sum is zero. If $\mathbf{x}(0)$ is a probability vector, then $\mathbf{p}$ is the stationary distribution and does not depend on the initial probabilities.