Stochastic matrix

A row (column) stochastic matrix is a square matrix each of whose rows (columns) consists of non-negative real numbers whose sum is equal to $1$.

Matrix $A=(a_{ij})$ is column stochastic if

• $a_{ij} \geq 0$ and
• $\sum\limits_i a_{ij} = 1.$

The Perron–Frobenius theorem implies that every stochastic (irreducible) matrix has an eigenvalue equal to $1$ and it's the largest absolute value of any eigenvalue.