Finding the inverse matrix of fourier transform

[tatianaiistb]

Homework Statement

If y=(1,0,0,0) and F₄*c=y, find c.

Homework Equations

c=F₄^-1*y

The Attempt at a Solution

I'm stuck. I don't know how to get F₄^-1.

F₄^-1 = (1/N) * [1, 1, 1, 1; 1 -i (-i)^2 (-i)^3; 1 (-i)^2 (-i)^4 (-i)^6; 1 (-i)^3 (-i)^6 (-i)^9] (this is a 4x4 matrix)

N = 4

So I'm confused because from that I get,

(1/4)* [1 1 1 1; 1 -i -1 i; 1 -1 1 -1; 1 i -1 -i]
(this is a 4x4 matrix)

I know that after I get this matrix, I just have to multiply by y to get c, but that inverse matrix has me confused.

Can anyone please help? Thanks!

 

[Mark44]

Homework Statement

If y=(1,0,0,0) and F₄*c=y, find c.

Homework Equations

c=F₄^-1*y

The Attempt at a Solution

I'm stuck. I don't know how to get F₄^-1.

F₄^-1 = (1/N) * [1, 1, 1, 1; 1 -i (-i)^2 (-i)^3; 1 (-i)^2 (-i)^4 (-i)^6; 1 (-i)^3 (-i)^6 (-i)^9] (this is a 4x4 matrix)

N = 4

So I'm confused because from that I get,

(1/4)* [1 1 1 1; 1 -i -1 i; 1 -1 1 -1; 1 i -1 -i]
(this is a 4x4 matrix)

Is what you have in the brackets above F₄?
If so, use it to find the inverse, F₄^-1.

I know that after I get this matrix, I just have to multiply by y to get c, but that inverse matrix has me confused.

Can anyone please help? Thanks!

 

[tatianaiistb]

I do not have F4... What I'm showing is what I've worked so far for inverse of F4...

 

[tatianaiistb]

I figured this out... I was actually doing it right and the inverse is that matrix I specified with the i's in it. The i's were throwing me off, but when you multiply that matrix by y, the i's cancel out and you can find c. :-)

 

[Mark44]

Your first post was not very clear on what you were given. Apparently you are given F₄^-1, so there's no need to find F₄.

F₄^-1 = (1/N) * [1, 1, 1, 1; 1 -i (-i)^2 (-i)^3; 1 (-i)^2 (-i)^4 (-i)^6; 1 (-i)^3 (-i)^6 (-i)^9] (this is a 4x4 matrix)

N = 4

and you simplified it to get
F₄^-1 = (1/4)* [1 1 1 1; 1 -i -1 i; 1 -1 1 -1; 1 i -1 -i][/quote]
You know y, and you have worked out that c = F₄^-1y, so just carry out the multiplication of your matrix and y.

$$c = \frac{1}{4}\begin{bmatrix}1&1&1&1\\1&-i&-1&i\\1&-1&1&-1\\1&i&-1&-1\end{bmatrix} \begin{bmatrix}1\\0\\0\\0\end{bmatrix}$$

 

[tatianaiistb]

Thank you!