  How could the Fourier and other transforms be naturally discovered if one
didn't know how to postulate them? In the case of the Discrete Fourier
Transform (DFT), we show how it arises naturally out of analysis of circulant
matrices. In particular, the DFT can be derived as the change of basis that
simultaneously diagonalizes all circulant matrices. In this way, the DFT arises
naturally from a linear algebra question about a set of matrices. Rather than
thinking of the DFT as a signal transform, it is more natural to think of it as
a single change of basis that renders an entire set of mutually-commuting
matrices into simple, diagonal forms. The DFT can then be "discovered" by
solving the eigenvalue/eigenvector problem for a special element in that set. A
brief outline is given of how this line of thinking can be generalized to
families of linear operators, leading to the discovery of the other common
Fourier-type transforms, as well as its connections with group representations
theory.
