 Solution

On a finite state space with at least two states, as in the surrounding questions, for a real function write $\pi (f)={\sum }_x\pi (x)f(x)$ and ${\mathrm{Var}}_{\pi }(f)={\sum }_x\pi (x)(f(x)-\pi (f){)}^2$. The Poincare inequality for a reversible Markov chain with constant $C$ iswhere the Dirichlet form of a Markov chain isThe associated spectral gap is the ordinary gap $\gamma =1-{\lambda }_2$, with variational formulaEquivalently minimize $\mathcal{E}(f,f)$ subject to $\pi (f)=0$ and $⟨f,f{⟩}_{\pi }=1$. Thus the Poincare inequality for a reversible Markov chain holds precisely for $C\ge 1/\gamma$, and the optimal constant is $1/\gamma$. It controls the ordinary rather than absolute spectral gap, so negative eigenvalues do not invalidate it. If the unspecified state space in this part is infinite, the Poincare inequality for a reversible Markov chain and the variational infimum still make sense on nonconstant functions in $L^2(\pi )$, using the self-adjoint operator on that Hilbert space; the resulting spectral gap need not be represented by an actual second eigenvalue. The finite-state formula above is the intended specialization.