[the_svd_doctor]

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.

[blt]

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