Solve Circulant Matrix Homework Equations

[Kosh11]

Homework Statement

http://i.imgur.com/ivZSA.png

Homework Equations

The Attempt at a Solution

This is my attempt http://i.imgur.com/ycuER.png

However I think I am doing something very wrong. My thought process is that to prove it is eigenvector is to apply X to the proposed eigenvector and see if I get a scalar multiple of itself. If so then by definition it's an eigenvector and that scalar would be the eigenvalue.

Edit: I now realize that the eigenvalue is (x1 + x2ζ + x3ζ^2 + x4ζ^3). Would it suffice for the proof to to multiply the eigenvector by the eigenvalue and show that equals X(eigenvector) where ζ^4 = 1?

 

[lanedance]

thought process sounds good, have a go

 

[lanedance]

ok, saw 2nd picture, now try taking a factors outside each component in the eigenvector, to show the length in each direction is proptional by the same factor (eigenvalue) to the original vector.
$$1 \ \xi \ \xi^2 \ and \ \xi^3$$

 

[lanedance]

you probably want to use some of the properties of $\xi$ as well, eg. $$\xi = e^{\frac{i n \pi}{2}}$$

 

[Kosh11]

Thanks I figured that part out. I have another question regarding part 2.

http://i.imgur.com/DnLXH.png

So I worked out the eigenvalues of that matrix to be 0,2,1-i,1+i. Since these are all unique eigenvalues there will be 4 linearly independent eigenvectors. But I'm not sure how to generalize that result to apply it to part 1.

 

[lanedance]

can;t see the pic, here's latex code for matrix, looks messy but is quite easy whe you get into it
$$\begin{pmatrix}<br /> x_1 & x_2 & x_3 & x_4 \\<br /> x_4 & x_1 & x_2 & x_3 \\<br /> x_3 & x_4 & x_1 & x_2 \\<br /> x_ 2& x_3 & x_4 & x_1 \\<br /> \end{pmatrix}$$