I've never given a second thought about what the etymology of "spectral" in spectral decomposition is. Somewhere in the back of my mind (and I guess many students of physics have the same notion) subconsciously i assumed it originates from eigenvalues of the Hamiltonian determining the atomic spectral lines . But I've never followed up on it and actually looked it up .

I might be wrong about the exact historical reason.

But the way I see it "spectral decomposition of A" is a way to express A as a sum of orthogonal, rank-1, operators. A = \sum l_i u_i u_i^T. Those l_i are the eigenvalues; u_i are the eigenvectors.

The eigenvectors look a whole lot like the "modes" in a Fourier decomposition. And if you plot (i, l_i), the eigenvalues are a bit like the "spectrum" (the amplitude of each mode).

In fact, the complex exponentials (the modes in the Fourier decomposition) are also eigenvectors of a specific operator (the Laplacian).

Math people are good at finding connections between things.

The spectrum of the matrix A is also closely related to the frequencies at which the ordinary differential equation xdot = Ax oscillates!

According to the almighty wikipedia, The connection is correct but it turned out to be an accident. David Hilbert who coined spectral theory was surprised when it was found to be applicable to solving quantum mechanical spectra.

If a spectrum is just a range of numbers, then the idea of spectrum should apply to many phenomenon. Many things can be described by a number. So, when matrix spectrum and atomic spectrum were formulated, perhaps spectrum was a word that was in vogue to describe a quantity. I think this is the explanation of the reason the same term is used for both. Because many connected phenomenon can be quantified. I know only a little about matrix spectrum and atomic spectrum, so take my thought with a grain of salt.

I love that on this website I can find comments that I simply don't understand a word of. Good on you for doing the stuff you do sir.