I don't know if this is nontrivial enough (or sufficiently unrelated a priori), but it is unusual not to see one often sees 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 illustrate this immediately with either simple (e.g. two-state) examples or various interesting Markov chains.
A standard statistical application of the fact that any positive operator has a unique square root is as follows. Suppose we have a column of data satisfying $\vec{y} = A \vec{x} + \vec{\epsilon}$, where the $\vec{\epsilon}$ have the correlation matrix $\mathbf{E}[\vec{\epsilon}^T \vec{\epsilon}] = \Omega$ (of course, therefore positive definite). When Then we can premultiply multiply on the equation left by $\Omega^{-1/2}$ and obtain (in transformed coordinates) a system with uncorrelated errors, which we can estimate using ordinary least squares. The resulting estimators therefore enjoy all the good properties of OLS. (See pages 2-3 of http://sociology.osu.edu/classes/soc703/Kaufman/Lec07_703.pdf)
These are not mathematically very high-powered applications, but ones hugely important in applied fields.
