= Solution
Take the real <inner product>
$$
\langle f,g\rangle_\pi=\sum_{x\in S}\pi(x)f(x)g(x).
$$
The <stationary distribution> is strictly positive for an <irreducible Markov chain>. <Detailed balance> gives $\langle Pf,g\rangle_\pi=\langle f,Pg\rangle_\pi$, so $P$ is a <self-adjoint operator>. Choose a real <orthonormal eigenbasis> $f_1,\ldots,f_m$, where $m=|S|$, $Pf_j=\lambda_jf_j$, $f_1\equiv1$, $\lambda_1=1$, and $\pi(f_j)=0$ for $j\geq2$.
For fixed $y$, apply the <spectral decomposition> to $g_y(z)=\mathbf1_{\{y\}}(z)/\pi(y)$. Its coefficient against $f_j$ is $f_j(y)$, whereas $(P^tg_y)(x)=P^t(x,y)/\pi(y)$. Thus the <spectral kernel expansion of a reversible chain> is
$$
\boxed{\frac{P^t(x,y)}{\pi(y)}=\sum_{j=1}^{m}\lambda_j^t f_j(x)f_j(y).}
$$
This holds for all integers $t\geq0$; at $t=0$ each multiplier is one, including when $\lambda_j=0$. Negative <eigenvalues> are allowed, and retain their signs for odd $t$.
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