# Fourier transform matrices

1. May 2, 2014

### jmomo

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.

2. May 2, 2014

### vela

Staff Emeritus
Does that mean anything to you? Because it doesn't to me.

3. May 2, 2014

### LCKurtz

This question should have been a continuation of this thread:

There the OP said he meant $5^{i\cdot j}$ instead of $5ij$. Dunno why he didn't correct it for this post.

4. May 2, 2014

### Zondrina

Might as well have continued this in your prior post. Now that you have $F$, the $i^{th}$ row of $\hat F$ has the form:

$$(1, \omega^{-i}, \omega^{-2i}, ..., \omega^{-(e-1)i})$$

Where $\omega$ is the e'th primitive root of unity. I'm sure you can continue.