Find eigenvalues and eigenvectors of particular Toeplitz matrix Assume a matrix in this form:
$$
\begin{bmatrix}
b       & c        & 0            & \dots        & 0          & a     \\
a       & b        & c            & 0            & \dots      & 0     \\
0       & a        & b            & c            & \ddots     & \vdots\\
\vdots  & \ddots   & \ddots       & \ddots       & \ddots.    & 0     \\
0       & \cdots   & 0            & a            & b          & c     \\
c       & 0        & \cdots       & 0            & a          & b     \\
\end{bmatrix}_{n \times n}
$$
I want to check that the eigenvalues are of the form
$$ \lambda_p = ae^{-2\pi ip/n}+b+ce^{2\pi ip/n}$$ with associated eigenvector $$ v_p = v_{p,j} 
= e^{-2\pi ipj /n}$$ I only find information about determinant of tridiagonal Toeplitz matrix but not for this case. Any help is appreciated!
 A: Simply multiply the matrix with each eigenvector and notice that you obtain
$$
\begin{bmatrix}
b       & c        & 0            & \dots        & 0          & a     \\
a       & b        & c            & 0            & \dots      & 0     \\
0       & a        & b            & c            & \ddots     & \vdots\\
\vdots  & \ddots   & \ddots       & \ddots       & \ddots.    & 0     \\
0       & \cdots   & 0            & a            & b          & c     \\
c       & 0        & \cdots       & 0            & a          & b     \\
\end{bmatrix}
v_p = (ae^{-2\pi ip/n}+b+ce^{2\pi ip/n})v_p = \lambda_p v_p
$$
directly from the matrix definition. It would be significantly harder to actually find $v_p$ and $\lambda_p$ without knowing them beforehand, but since you are given $n$ candidate solutions (with $1 \leq p \leq n$), one only needs to check that they indeed form eigenpairs.
A: In the case you don't know beforehand the eigenvectors, you can refer to the fact that your matrix is circulant. The eigenvectors of a $n \times n$ circulant matrix are the columns of the Discrete Fourier matrix $F_n$ (this is also explained in the referenced Wikipedia article) and it is precisely those which are given to you...
