Background: While studying (finite) Markov chains I came across a characterization, different than the one using a transition matrix $p_i(j)$. Namely, we can specify the "diagram" with probabilities $p(i,j)$, i.e., the relative frequency of the transitions $p_i(j)$.
Denoting the probability of staying at the state $i$ by $p(i)$ we get the following relations between these three quantities1:
- $p(i) = \sum_j p(i,j) = \sum_j p(j,i) = \sum_j p(j)p_j(i)$
- $p(i,j) = p(i)p_i(j)$
- $\sum_j p_i(j) = \sum_i p(i) = \sum_{i,j} p(i,j) = 1$.
In particular, the relations of the first bullet point imply that for the matrix $P = (p(i,j)_{i,j})$ the row sums are equal to the column sums (but not necessarily constant).
Question: Do matrices, with the property of having equal row sums and column sums have a particular name? Is this a well-studied2 class of matrices? I can see that symmetric matrices and double stochastic matrices are special cases thereof (with double stochastic matrices probably being closer to the spirit of what I am looking for), and I was wondering of there is a known generalization of both of these concepts.
1 See, for example, page 7 in A Mathematical Theory of Communication.
2 I found related questions here.
