Let $v$ be a (fixed) unit vector in $\mathbb { R } ^ { 3 }$. (We think of elements of $\mathbb { R } ^ { n }$ as column vectors.) Let $M = I _ { 3 } - 2 v v ^ { t }$. Pick the correct statement(s) from below.
(A) $O$ is an eigenvalue of $M$.
(B) $M ^ { 2 } = I _ { 3 }$.
(C) 1 is an eigenvalue of $M$.
(D) The eigenspace for the eigenvalue $-1$ is 2-dimensional.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
What can be said about matrices whose eigenvalue circle is tangent to the $x$-axis and whose center is located on the $y$-axis?

Determine the spectrum of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and a basis of eigenvectors of $B$.

Determine the set of eigenvalues of $A _ { n }$ and a basis of eigenvectors.

Show that every circulant matrix is diagonalizable. Specify its eigenvalues and a basis of eigenvectors.

For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Deduce that $J_n^{(1)}$ admits the following eigenvalues, enumerated with their multiplicity: $$\lambda_k = 2\cos\left(\frac{2\pi k}{n}\right), \quad k \in \llbracket 0, n-1 \rrbracket.$$