> [!NOTE] Definition > Let $S$ be a countable set. > A square matrix $A=\left(A_{i j}\right)_{|S| \times|S|}$ is called substochastic if > > $0 \leq A_{i j} \leq 1, \quad i, j \in S \quad \text { and } \quad \sum_{j \in S} A_{i j} \leq 1, \quad i \in S$ that is, the sum of each row is at most $1$. > > > A square matrix $A=\left(A_{i j}\right)_{|S| \times|S|}$ is called stochastic if > > $0 \leq A_{i j} \leq 1, \quad i, j \in S \quad \text { and } \quad \sum_{j \in S} A_{i j}=1, \quad i \in S$ that is, the sum of each row equals $1$.