Explicit solution to the invariant distribution of a Markov chain

Let $$\{X_t\}_t$$ be a discrete-time Markov chain with right stochastic transition matrix $$P$$ and a unique invariant distribution $$\pi$$. Let the state space be $$\{1,\dots,S\}$$. Is there an explicit solution (potentially an approximation) for the element $$\pi^{(s)}$$ of $$\pi$$ in terms of the elements of $$P$$ for a given $$S$$?

For instance, if $$\begin{equation} P = \begin{bmatrix} 1-\alpha, \alpha \\ \beta, 1-\beta \end{bmatrix} \end{equation}$$ then $$\begin{equation} \pi = \begin{bmatrix} \frac{\beta}{\alpha+\beta} \\ \frac{\alpha}{\alpha+\beta} \end{bmatrix}, \end{equation}$$ and I am interested in the generalization for $$S > 2$$.

For ergodic $$\mathbf{P},$$ you seek $$S$$-vector$$\pi$$ with $$\pi\mathbf{P}=\pi.$$ So $$\pi$$ is a left eigen-vector of $$\mathbf{P}.$$ You can use R to compute $$\pi.$$

Illustration:

P = matrix(c(.1, .2, .7,
.2, .7, .1,
.7, .1, .2), byrow=T, nrow=3)

P
[,1] [,2] [,3]
[1,]  0.1  0.2  0.7
[2,]  0.2  0.7  0.1
[3,]  0.7  0.1  0.2

g = eigen(t(P))$vec[,1] g = as.numeric(g) pi = g/sum(g); pi  0.3333333 0.3333333 0.3333333 pi %*% P [,1] [,2] [,3] [1,] 0.3333333 0.3333333 0.3333333 I used a doubly-stochastic matrix so the answer would be obvious, but the method works generally. The invariant (stationary) vector is proportional to the left eigen vector of smallest modulus. Notes on R code: (a) Use transpose t(P) because R finds right eigen-vectors. (b) Use$vec[,1] to get the first column of the display of eigen vectors; R prints the eigen-vector of smallest modulus first.

(c) Use as.numeric to suppress (possible) complex-number notation. [For an ergodic $$\mathsf{P},$$ the first eigen-vector is always real, but other vectors in the display may be complex.]

(d) Use pi = g/sum(g) to make the elements of $$\pi$$ sum to $$1.$$

(e) The last line is to verify that $$\pi$$ is the desired vector. (Unnecessary, but guards against typos in the code.)