Markov chain transition matrix

Question

if $P$ and $Q$ are $n \times n$ transition matrices for two Markov chain, then product $R=PQ$ is also a transition matrix.

is this true ? why is it ?

looks like product of transition matrix means transition $P$ and $Q$.

Answer 1

If $P$ and $Q$ are $n\times n$ matrices s.t.

$$\sum_{j=1}^n P_{ij}=1,$$

$$\sum_{j=1}^n Q_{ij}=1,$$

for all $i=1,\dots,n$, then $R=PQ$ satisfies the same property.

Explicitly $$\sum_{j=1}^n R_{ij}= \sum_{j=1}^n\left( \sum_{k=1}^n P_{ik}Q_{kj}\right)=\sum_{k=1}^n P_{ik}\left(\sum_{j=1}^n Q_{kj} \right)=\sum_{k=1}^n P_{ik}\cdot 1=1,$$

for any $i=1,\dots,n$.