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I don't know if this is nontrivial enoug (or sufficiently unrelated a priori), but it is unusual not to see the spectral theorem in proofs that any aperiodic, irreducible Markov chain with finitely many states converges to its stationary distribution. The fact that the spectral gap measures convergence speed is also an immediate an rewarding consequence, and one can see this immediately with either simple (e.g. two-state) examples or various interesting Markov chains.