Show norm preserving property and determine Eigenvalues Can someone of you give me a solution for this?

Let $N\in \mathbb N$.
  a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{1}{\sqrt N}\sum_{j=1}^Nx_je^{2\pi i\frac{(j-1)(k-1)}{N}} \quad \forall k\in \{1,...,N\}$$ 
  Show that $\mathfrak F$ is norm-preserving, i.e. $$||(\mathfrak F(x))_k||_2 = ||x||_2 \quad \forall k\in \{1,...,N\}$$ 
  b) For $n,m \in \{1,2,...,N\}$ we define the entries of $M\in \mathbb C^{N\times N}$ by $$M_{nm} := 
\begin{cases}
\frac{N+1}{2}, \quad \quad n=m \\
\frac{1}{e^{2\pi i \frac{m-n}{N}}-1}, \,\,n \neq m
\end{cases}$$
  Show that $1, 2, 3, ..., N$ are the Eigenvalues of M.

 A: a), Like Omnomnomnom's opinion, the DFT matrix shows a good guidance. Here, DFT matrix $\mathsf{W}$ is defined as below:
$$
\mathsf{W}=\cfrac{1}{\sqrt{N}}
\begin{bmatrix}
\omega_{1,1} & \omega_{2,1} & \omega_{3,1} & \cdots & \omega_{N,1} \\
\omega_{1,2} & \omega_{2,2} & \omega_{3,2} & \cdots & \omega_{N,2} \\ 
\vdots       & \ddots       & \ddots       & \ddots & \vdots \\
\vdots       & \ddots       & \omega_{j,k} & \ddots & \vdots \\
\vdots       & \ddots       & \ddots       & \ddots & \vdots \\
\omega_{1,N} & \cdots       & \cdots       & \cdots & \omega_{N,N}
\end{bmatrix}
$$
where,
$$
\omega_{j,k}=\exp\biggl(2\pi i\cfrac{(j-1)(k-1)}{N}\biggr).
$$
This matrix is an orthogonal matrix, and has a property that the calculation $\mathsf{W}^{\text{T}}\mathsf{W}$ shows just an identity matrix. Therefore the mapping can be described by DFT matrix and the following vector:
$$
\boldsymbol{x}=
\begin{bmatrix}
x_{1} & x_{2} & \cdots & x_{j} & \cdots & x_{N}
\end{bmatrix}
^{\text{T}}
$$
so that:
$$
\begin{aligned}
(\mathfrak F(\boldsymbol{x})) =& 
\begin{bmatrix}
(\mathfrak F(x))_{1} & 
(\mathfrak F(x))_{2} & \cdots &
(\mathfrak F(x))_{k} & \cdots &
(\mathfrak F(x))_{N}
\end{bmatrix}
^{\text{T}}\\[5pt]=& \ 
\mathsf{W}\boldsymbol{x}
\end{aligned}
$$
This matrix multiplication denotes the Discrete Fourier Transform(DFT) of the signal $\boldsymbol{x}$. Owing to the above description, the norm-preserving $||(\mathfrak F(x))_k||_2$ can be proved as follows:
$$
\begin{aligned}
||(\mathfrak F(x))_k||_2^2 =& \sum_{k=1}^{N} (\mathfrak F(x))_{k}^2 \\=&
(\mathfrak F(\boldsymbol{x}))^{\text{T}}(\mathfrak F(\boldsymbol{x})) \\=&
(\mathsf{W}\boldsymbol{x})^{\text{T}}(\mathsf{W}\boldsymbol{x}) \\=&
\|\boldsymbol{x}\|_{2}^2
\end{aligned}
$$

b), At first, the matrix $\mathsf{M}$ can be expressed as below:
$$
\mathsf{M}=
\begin{bmatrix}
m_{0} & m_{-1} & m_{-2} & \cdots & m_{1-N} \\
m_{1} & m_{0}  & m_{-1} & \cdots & m_{2-N} \\ 
m_{2} & m_{1}  & m_{0}  & \cdots & m_{3-N} \\ 
\vdots       & \ddots   & \ddots  & \ddots & \vdots \\
\vdots       & \ddots   & m_{k}   & \ddots & \vdots \\
\vdots       & \ddots   & \ddots  & \ddots & \vdots \\
m_{N-1}      & \cdots   & \cdots  & \cdots & m_{0}
\end{bmatrix}
$$
where $k=m-n$ and:
$$
m_{k}=
\begin{cases}
\cfrac{N-1}{2} & (k=0)\\
\cfrac{1}{\exp\biggl(2\pi i\cfrac{k}{N}\biggr)-1} & (\text{otherwise})
\end{cases}
$$
Of course, it is also established as $m_{N-k}=m_{-k}$. Then, we must be noticed that the matrix of arrangement has a particular pattern. These matrices are called circulant matrix. In generally, circulant matrix has a very interesting and beautiful property that eigenvalues $\lambda_{j}$ can be determined uniquely like follows:
$$
\begin{aligned}
\lambda_{j}=m_{0}+
\sum_{k=1}^{N-1}m_{-k}\omega_{j,k}
\end{aligned}
$$
In this case, these eigenvalues are follows:
$$
\begin{aligned}
\lambda_{j}=
\cfrac{N-1}{2}+
\sum_{k=1}^{N-1}\cfrac{\omega_{j,k+1}}{1-\omega_{1,k}}
\end{aligned}
$$
