Real time signal processing use cases for eigenvalues of symmetric matrices
OK, you know that complex sinusoids are eigenfunctions (eigenvectors) of linear time-invariant (LTI) systems (which are linear operators). Guess what you see when you look at a spectrum: it's the eigenvalues corresponding to the eigenvector of any given frequency. So, basic signal theory uses eigenvectors to understand the fact that the frequency- and time-domain representation of signals are equivalent.

In MUSIC especially, the eigenvalues are of great importance – they signify the power of that eigenvector of the observed system (i.e. the power of the frequency component in case of that eigenvector spanning the signal subspace). So, MUSIC depends on the eigenvalues.

Basically, all symmetrical MIMO schemes that involve precoding decompose a matrix using the EVD into especially a diagonal matrix with the Eigenvalues as entries – and the question whether, and with what power, a decomposed "virtual" channel is used depends on the magnitude of the respective eigenvalue. So, MIMO, Precoding, and in the context of power assignment waterfilling for these channels depend on eigenvalues.



Now, for things that are "borderline" signal processing:

Graph theory has a relatively rich interaction with linear algebra. The Spectrum of a Graph is basically equivalently used to Spectra of Matrices. 

Especially for random graphs, statements on bounds and distribution of eigenvalues of the (often unboundedly large) adjacency matrix are useful to describe degree distributions, and how reliably one can remove random edges from the graph without cutting the graph.

You can imagine the relevance of that to the theory of routing in communication networks; but in the case of random parity-check codes, the distribution of node degrees and cycle lengths is an important thing that can be at lest approximated through this.