[enchiridion]

Do you mean the FFT? I've been trying to wrap my head around the Fourier transform lately, and I can see the matrix connection to an FFT, but not the Fourier transform in general.

[bonoboTP]

The Fourier transform performs a change of basis into the eigenbasis of convolution.

It's linear and so in the discrete case can be expressed as matrix multiplication (every discrete linear operation is expressible as matmul).