There are a couple of other comments that have mentioned oscillation modes, vibrations, etc. The first 7 pages of this series on sound synthesis might help give

There are a couple of other comments that have mentioned oscillation modes, vibrations, etc. The first 7 pages of this series on sound synthesis might help give an idea of where these might come from: https://drive.google.com/file/d/12SM0SAOvMq166gc8B1b81Y_S7HPym3Iy/view https://drive.google.com/file/d/12SM0SAOvMq166gc8B1b81Y_S7HP... The third page in particular shows a plot of "amplitude" versus "frequency" to show the "harmonic spectrum of a sawtooth wave". The "frequencies" correspond to the modes of vibration (i.e., sine waves of different frequency), which are the "eigenvectors" in this case. The "amplitudes" are the relative contribution of those vibrations to the overall sound, and these correspond to "eigenvalues". The article is talking purely about constructing sounds via synthesis, so there's not necessarily a linear system associated with it, but there is a connection. Wave equations represented by linear partial differential equations can often be analyzed as a linear system that has these "modes of vibration" (i.e., series of orthogonal sinusoids at different frequencies). If you were to, for example, model a plucked string (like a guitar), you can model the solution as a weighted sum of eigenvectors (in this case, "modes of vibration" or sinusoids of different frequencies). The "weights" would be the eigenvalues, which determine the spectrum and ultimately the timbre of the sound produced. That might seem more involved, because it's an infinite-dimensional linear system (i.e., the vectors are functions on a interval, rather than finite lists of numbers). It turns out, though, that the finite-dimensional discretization of an infinite-dimensional linear system (i.e., a partial-differential equation approximated by a finite-dimensional linear system) will sometimes have eigenvectors / eigenvalues that have similar features as the infinite-dimensional case. For example, there are certain finite-difference operators that can be written in matrix form whose eigenvectors will work out to be sampled sinusoids. I'm not totally sure of the history, but I think a lot of the interest in eigenvectors / eigenvalues as a topic in matrix theory originated from this are (i.e., numerical solutions for partial-differential equations that were used to model physical systems).