Fast Fourier Transform using a Vandermonde Matrix - Evaluation of Co-efficients?

Question

Say i'm trying to evaluate the Polynomial:

x^2 + 1

Using the Fast Fourier transform method for evaluating co-efficients. Now i can change this into matrix/vector form using the co-effcient as inputs for the fast fourier transform:

so:

x^2 + 1 = <1, 0, 1, 0>

This is done by using the coefficient value e.g 1 = 1, 0x^1 = 0, X^2 = 1 and so on

Now we get to the bit where i'm totally confused. I'm meant to use the vandermonde matrix :Vandermonde matrix ~ Wiki to evaluate these values into FFT Form using the matrix:

1 1 1 1  
1 i-1-i
1-1 1-i
1-i 1 i

The output of

fft(1,0,1,0)

is

(2,0,2,0)

Now thats the step i don't quite understand, how did we use that matrix to get (2,0,2,0)?

Answer 1

First, your Vandermonde matrix is incorrect. The (4,3) entry should be -1, not 1, since the fourth row should be (-i)⁰, (-i)¹, (-i)², (-i)³. Note in particular that

(-i)*(-i) = (-1)² * i² = i² = -1.

With this correction, the result follows from multiplying the Vandermonde matrix by the column vector (1,0,1,0).