The discrete "Fast Fourier Transform", (FFT) is based on the
factorization of the Fourier matrix.  If n = 8, we have
the 8x8 Fourier matrix, which is this:


        1    1    1    1    1    1    1    1
        1    w    w^2  w^3  w^4  w^5  w^6  w^7
        1    w^2  w^4  w^6  w^8  w^10 w^12 w^14
        1    w^3  w^6  w^9  w^12 w^15 w^18 w^21
F  =    1    w^4  w^8  w^12 w^16 w^20 w^24 w^28
        1    w^5  w^10 w^15 w^20 w^25 w^30 w^35
        1    w^6  w^12 w^18 w^24 w^30 w^36 w^42
        1    w^7  w^14 w^21 w^28 w^35 w^42 w^49


where w = e^(2pi i/8), a primitive complex eighth root
of unity

We first permute the columns in such a way that all even
columns precede all odd columns.  This is acocmplished
by postmultiplying F by the permutation matrix P, where


        1    0    0    0    0    0    0    0
        0    0    0    0    1    0    0    0
        0    1    0    0    0    0    0    0
        0    0    0    0    0    1    0    0
        0    0    1    0    0    0    0    0
        0    0    0    0    0    0    1    0
        0    0    0    1    0    0    0    0
        0    0    0    0    0    0    0    1


The 4x4 submatrix in the upper left corner of F is then this

        1    1    1    1   
        1    w^2  w^4  w^6
f  =    1    w^4  w^8  w^12
        1    w^6  w^12 w^18


Since w^2 = e^(2pi i/4), f is seen to be the 4x4 Fourier matrix.

Furthermore, since w^8 =1, the 4x4 submatrix in the lower
left corner of F is seen to be another copy of f.  The
4x4 submatrix in the upper right corner of F is

        1    1    1    1   
        w    w^3  w^5  w^7
        w^2  w^6  w^10 w^14
        w^3  w^8  w^15 w^21

This is equal to the product of f premultiplied by the
appropriate diagonal matrix,


        1    0    0    0   
        0    w    0    0
        0    0    w^2  0   
        0    0    0    w^3  

In other words, we have




        f   Df      I  D     f   0
FP =    f  -Df  =   I -D     0   f



f, in turn, can be similarly factored.  In general, if n = 2^k,
the nxn Fourier matrix can be factored into a product of three
matrices.  The first is of the form  


       I    D
       I   -D


where D is diagonal, with consecutive powers of w on its diagonal.

The second is of the form

       f   0
       0   f


where f is the n/2 by n/2 Fourier matrix, and the last is
a permutation matrix that moves the even columns before the odd columns.