Positive matrices and nonnegative matrices

notes

## Stochastic Matrix

A matrix is (lefty) stochastic/Markov, if it is nonnegative and its column sums are 1.

Theorem: If $A$ is Markov, then TFAE (the followings are equivalent):

• $\lambda_{1} = 1$;
• Eigenvector $\mathbf{x}_{1}$ is non-negative;
• $| \lambda_{i} | \leq 1, \forall i \ne 1$;
• If any power of $A$ is positive, then $\vert \lambda_{i} \vert \leq 1, \forall i \ne 1$, and $A^{k} \mathbf{u}_0 \to c \mathbf{x}_1$, when $k \to \infty$.