# 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
Show that $\mathfrak F$ is norm-preserving, i.e.
b) For $n,m \in \{1,2,...,N\}$ we define the entries of $M\in \mathbb C^{N\times N}$ by
Show that $1, 2, 3, ..., N$ are the Eigenvalues of M.

a), Like Omnomnomnom’s opinion, the DFT matrix shows a good guidance. Here, DFT matrix $\mathsf{W}$ is defined as below:

where,

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:

so that:

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:

b), At first, the matrix $\mathsf{M}$ can be expressed as below:

where $k=m-n$ and:

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:

In this case, these eigenvalues are follows: