Stochastic matrix with structure

Question

Let $P \in [0,1]^{(n \times n)}$ be a stochastic matrix i.e $P_{ij} > 0 ~ \forall i,j$ and $\sum_{j}P_{ij} = 1~ \forall i$. Now let us impose additional structure on $P$ by saying that $P_{ij} + P_{ji} = \frac{1}{n} ~\forall i \neq j$ (*). Do these matrices have a special name in literature? Are their properties well studied? Any references would be appreciated. Thanks!

(*) - Edited

Answer 1

Let $P$ be as described. Then

$$\sum_i \sum_j P_{ij} = \sum_i 1 = n,$$

but

\begin{align*} \sum_j \sum_i P_{ij} &= \sum_j \left( P_{jj} + \sum_{i,\ i \ne j} (1 - P_{ji}) \right) = \sum_j \left( P_{jj} - (1 - P_{jj}) + \sum_i (1 - P_{ji}) \right) \\ &= \sum_j \left( 2P_{jj} - 1 + \sum_i (1 - P_{ji}) \right) = \sum_j \left( 2P_{jj} - 1 + n - \sum_i P_{ji} \right) \\ &= \sum_j \left( 2P_{jj} - 1 + n - 1 \right) = 2 \sum_j P_{jj} - 2n + n^2. \end{align*}

Both of these sums represent sum of all the elements in the matrix, so

$$n = 2 \sum_j P_{jj} - 2n + n^2,$$

i.e.,

$$n^2 - 3n + 2 \sum_j P_{jj} = 0.$$

Denoting $s := \sum_j P_{jj}$, we see that

$$n = \frac{3 \pm \sqrt{9 - 8s}}{2} \le 3. \tag{*}$$

So, your matrices must be of order:

• $n = 1$: the only is $P = \begin{bmatrix} 1 \end{bmatrix}$;
• $n = 2$: all have the form $P = \begin{bmatrix} 1-x & x \\ 1-x & x \end{bmatrix}, \ x \in [0,1]$; or

• $n = 3$: all have the form $P = \begin{bmatrix} 0 & x & 1-x \\ 1-x & 0 & x \\ x & 1-x & 0 \end{bmatrix}, \ x \in [0,1]$.

  The diagonal zeroes come from $(*)$. As Did wrote in the comments, such $P$ is the transition matrix of a biased random walk on the cycle $\mathbb{Z}/3\mathbb{Z}$.

I expect there is no name for them, nor they are much researched, since there is not really that many of them.