= Spectral kernel expansion of a reversible chain
{title2=$P^t(x,y)/\pi(y)$}
For a finite <irreducible Markov chain> satisfying <detailed balance> with $\pi$, its <transition matrix> is a <self-adjoint operator> for $\langle f,g\rangle_\pi=\sum_x\pi(x)f(x)g(x)$. For a real <orthonormal eigenbasis> $(f_j)$ with <eigenvalues> $(\lambda_j)$,
$$
\frac{P^t(x,y)}{\pi(y)}=\sum_j\lambda_j^t f_j(x)f_j(y).
$$
Indeed, apply the <spectral decomposition> of $P^t$ to the function $\mathbf1_{\{y\}}/\pi(y)$, whose coefficient against $f_j$ is $f_j(y)$. For $t=0$, every <eigenvalue> multiplier is interpreted as one.
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