# Find eigenvalue via Discrete Fourier Transform

The question is as following

Let $B=\begin{bmatrix} 4 & 1 & 0 & 0 & 1 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 \\ 1 & 0 & 0 & 1 & 4 \\ \end{bmatrix}$ and $x=\begin{bmatrix} cos(\theta + \phi) \\ cos(2\theta + \phi) \\ cos(3\theta + \phi) \\ cos(4\theta + \phi) \\ cos(5\theta + \phi) \\ \end{bmatrix}$
a) $\exists a,b,c \in R$ such that $Bx - ax = x=\begin{bmatrix} b \\ 0 \\ 0 \\ 0 \\ c \\ \end{bmatrix}$, where $(a,b,c)=?$
(please express a, b, and c in terms of $\theta$ and $\phi$)

b) All the eigenvalues of B are?

One of my classmate said that (b) could be solved by Discrete Fourier Transform; however, I still have no idea how to approach the problem by it. And (a) still confuses me by those symbols.

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Since $A$ is a convolution, conjugation with the Discrete Fourier Transform $\mathcal{F} A \mathcal{F}^{-1}$ will diagonalize $A$. – WimC Jan 20 '13 at 17:01
@lucasKoFromTW: Did you write $a)$ correctly? Hint for b: Do you notice that the matrix $B$ is symmetric? From this, what sort of matrix must it be and what can you say about its eigenvalues from this knowledge? Do you also know its definiteness (positive or negative)? Additional hint, do you see that $B$ is a Circulant Matrix? How does this help you use your friend's hint? Regards – Amzoti Jan 20 '13 at 17:37