Given a Markov chain with a stationary distribution $\mathbf{\pi}$ and a transition matrix P, it satisfies the Detailed Balance Condition if:
$$ \pi_i P_{ij}= \pi_j P_{ji}, \quad \forall i,j $$
My question is, how to derive this equation?
I have tried the 2-dimensional version:
Given: $$ \pi=[\pi_1,\pi_2] $$
$$ P= \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} $$
When stationary:
$$ \pi P = \pi $$
Then:
$$ \pi_1 P_{11} +\pi P_{21} =\pi_1 \\ \pi_1 P_{12} +\pi P_{22} =\pi_2 $$
The two equaitons can be simplified to the same equation:
$$ \pi_1 P_{12} =\pi_2 P_{21} $$
And this conforms to the Detailed Balance Equation.
But how to prove the 3-dimensional version and the general n-dimensional version?