This is a question continued from here... Proof about Steady-State distribution of a Markov chain
I have a stochastic matrix $P_\delta$ of dimensions $n\times n$. I look at the matrix as a transition matrix of some Markov chain $M$. Matrix $P_\delta$ is dependent of $\delta\in(0,1]$. For this $P_\delta$ I would like to compute the steady state distribution $\pi_\delta$.
I am interested in elements of $\pi_\delta$ with respect to $\delta$. From my experiments I concluded, that for each $i$, the $\pi_{\delta}(i)$ is monotonic function of $\delta$ (either increasing or decreasing).
So for some $i\in[1,n]$ I can write a formula for $\pi_\delta(i)$:
$$ \pi_\delta(i)=\sum_{j=1}^n \pi_\delta(j) \frac{k(j)}{1+\delta k(j)} (a(j,i)+\delta I(j,i)) $$
Here $k(j)\in [1,10]$, $a(j,i)\in[1,10]$ and $I(j,i)\in\{0,1\}$ are constants. Also note that the right-hand side of the equation includes $\pi_\delta(i)$.
How can I conclude that for each i, the $\pi_\delta(i)$ is monotonic function of $\delta$? I think it is not enough to derive the right side by $\delta$ and show that sign of the derivative does not change, because $\pi_\delta(j)$ are dependent of $\delta$.
