I'm trying to understand how the equilibrium distribution satisfy the detailed balance equation.
To my understanding, I only understand that a detailed balance equation would only be satisfied if $\pi_j$ $p_{j,k}$ = $\pi_k$ $p_{k,j}$
Say I have a very simple Markov Chain with states {1,2,3} which has the following transition matrix.
$$P= \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\\\end{bmatrix}$$
If I have a equilibrium distribution of $\pi_1$ = $\pi_2$ = $\pi_3$ of 1/3 each, how does the equilibrium distribution satisfy the detailed balance equations? - any helps would be highly appreciated!
For an arbitrary markov chain, having an equilibrium distribution doesn't imply satisfying the detailed balance equations. The example you have given is an example of a chain with an equilibrium distribution not satisfying detailed balance.
The converse is true though, if a chain satisfies detailed balance with a function $f_i$ so $f_i p_{ij} = f_j p_{ji}$ then it has an equilibrium distribution $\pi_{i}=f_{i}$