Fourier transform matrices

1. The problem statement, all variables and given/known data

Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F_13 for i, j = 0,1,2, 3.

Compute F(hat) and verify that F(hat)F = I





2. Relevant equations

The matrix F(hat) is called the inverse discrete Fourier transform of F.





3. The attempt at a solution

I found that e = 4, so (F)F(hat) = 4 I, so F(1/4 F(hat)) = I

I calculated that matrix F=

1 1 1 1

1 5 12 8

1 12 8 1

1 8 1 5



My Question: How do I calculate matrix F(hat)? I understand it is the inverse of F, but I am unsure of how to calculate it.