The text focusses on general square matrices. In class, we simplify the narrative by not worrying about general matrices, focussing instead only on symmetric matrices.

We motivated eigenvalues with two demos: the best line that captures the scattering of points and the eigenfaces example. At this level, thinking of eigenvalues as capturing the importance of different aspects of an operator is a good approach. It is also important to keep in mind that spectral decomposition is a property of linear systems in general, not just matrices.

In fact, the entirety of EE 315 is representation of functions using an eigenbasis formed by the eigenfunctions of the derivative operator. The parallels can be made formally exact (though with significant technical effort and abstractions).