I just have a quick question about Markov Chain and linear algebra.
Background. Let $\{M_n: n= 0, 1, 2, \dots \}$ be a Markov Chain. We can represent the transition probabilities $_{n}Q^{(i,j)}$ in a $s \times s$ matrix $Q$. Note that $_{n}Q^{(i,j)}$ is the same thing as $P(M_{m+n} = j|M_{m} = i)$.
Question. Is the matrix $Q$ the same in the sense of linear transformations? In particular, suppose we have two vector spaces $V$ and $W$. If we choose an ordered basis for $V$ and an ordered basis for $W$ we can represent linear transformations from $V$ to $W$ as matrices. So then we can talk about the rank, nullity, kernel, etc...But does the term matrix in context of Markov chains just mean an array of numbers?