We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
Let $\Phi$ be the matrix in the basis $(f_1, f_2, f_3)$ of the endomorphism $\varphi$ of $F$ induced by $c$ (as determined in I.B).
In this question, we propose to calculate the spectrum of $\Phi$ without calculating its characteristic polynomial. I.C.1) Why is 1 an eigenvalue of $\Phi$? I.C.2) Can we deduce from the sole calculation of the trace of $\Phi$ that $\Phi$ is diagonalizable in $\mathscr{M}_3(\mathbb{C})$? I.C.3) Calculate $\Phi^2$. Using the additional information obtained by calculating the trace of $\Phi^2$, determine the spectrum of $\Phi$. Is the matrix $\Phi$ diagonalizable in $\mathscr{M}_3(\mathbb{R})$?

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
I.D.1) Deduce from the previous questions the spectrum of $C$. Specify the multiplicity order of the eigenvalues. I.D.2) Is the matrix $C$ diagonalizable over $\mathbb{C}$? over $\mathbb{R}$? If yes, indicate a diagonal matrix similar to $C$.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Demonstrate that $H_n$ admits $n$ real eigenvalues (counted with their multiplicity) that are strictly positive.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that $U$ admits eigenvalues in $\mathbb{C}$, that they are real and that two eigenvectors associated with distinct eigenvalues are orthogonal.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$, what do the solutions of the equation $\varphi_A(x,0) = 0$ represent for $A$? Specify the number of real eigenvalues of $A$ according to the value of $\Delta_A = (a-d)^2 + 4bc$.

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. According to the values of $(\alpha, \beta, \gamma)$ (where $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$), specify the number of real eigenvalues of $A$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. Let $U$ be one of the points of $\mathcal{C}(\Omega, r)$ at which the tangent line contains $K$.
Express the eigenvalues of $A$, considered as an element of $\mathcal{M}_2(\mathbb{C})$, using the abscissa of $K$ and the distance $KU$ from $K$ to $U$.

In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question, $\Omega = (\alpha, \beta) \in \mathbb{R} \times \mathbb{R}^*$, $r = |\beta|$ and $E = (\alpha + |\beta|, \beta)$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$.

In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question $\Omega = (0, \alpha)$ with $\alpha > 0$ and $r = \alpha/2$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$. Make a drawing in the case where $\alpha = 6$ illustrating questions IV.C.2 and IV.C.3.

Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Calculate the point spectrum of $S$ and $V$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the point spectrum of $S$ and $V$ in $F$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the spectrum of $S$ and $V$ in $F$.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
Show that if $n$ is odd, then $f$ admits at least one real eigenvalue.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
In this question, $\lambda = \alpha + \mathrm{i}\beta$, with $(\alpha, \beta)$ in $\mathbb{R}^2$, is a non-real eigenvalue of $M$ and $Z$ in $\mathcal{M}_{n,1}(\mathbb{C})$, non-zero, is such that $MZ = \lambda Z$.
If $M = (m_{i,j})_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ we denote $\bar{M} = (m_{i,j}')_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ with $m_{i,j}' = \bar{m}_{i,j}$ (conjugate of the complex number $m_{i,j}$) for all $(i,j)$ in $\llbracket 1, n \rrbracket^2$ and if $Z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}$ we denote $\bar{Z} = \begin{pmatrix} z_1' \\ \vdots \\ z_n' \end{pmatrix}$ with $z_i' = \bar{z}_i$ for all $i$ in $\llbracket 1, n \rrbracket$.
We set $X = \frac{1}{2}(Z + \bar{Z})$ and $Y = \frac{1}{2\mathrm{i}}(Z - \bar{Z})$.
IV.C.1) Verify that $X$ and $Y$ are in $E$ and show that the family $(X, Y)$ is free in $E$.
IV.C.2) Show that the vector plane $F$ generated by $X$ and $Y$ is stable by $f$ and give the matrix of $f_F$ in the basis $(X, Y)$.

Specify the set of eigenvalues of $\tau$. Is the application $\tau$ diagonalizable?

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that (II.2) admits a nonzero periodic solution of period $p$ if and only if 1 is an eigenvalue of $Q$.

Using the results of Q30 and Q31, deduce that there exists $j \in \llbracket 1, q \rrbracket$ such that $\lambda = 2\cos\frac{j\pi}{q+1}$.

Let $V = {}^{ t } \left( v _ { 1 } , \ldots , v _ { n } \right)$ be an eigenvector of $A _ { n }$ associated with a complex eigenvalue $\lambda$, where $A_n$ is the square matrix of size $n$:
$$A _ { n } = \left( \begin{array} { c c c c c c } 2 & - 1 & 0 & \ldots & \ldots & 0 \\ - 1 & 2 & - 1 & \ddots & & \vdots \\ 0 & - 1 & 2 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 2 & - 1 \\ 0 & \ldots & \ldots & 0 & - 1 & 2 \end{array} \right)$$
Show that $\lambda$ is necessarily real and that the components $v _ { i }$ of $V$ satisfy the relation:
$$v _ { i + 1 } - ( 2 - \lambda ) v _ { i } + v _ { i - 1 } = 0, \quad 1 \leq i \leq n$$
where we set $v _ { 0 } = v _ { n + 1 } = 0$.

Let $\lambda$ be an eigenvalue of $A _ { n }$.
(a) Show that the complex roots $r _ { 1 } , r _ { 2 }$ of the polynomial
$$P ( r ) = r ^ { 2 } - ( 2 - \lambda ) r + 1$$
are distinct and conjugate.
(b) We set $r _ { 1 } = \overline { r _ { 2 } } = \rho e ^ { i \theta }$ with $\rho > 0$ and $\theta \in \mathbb { R }$.
Show that we necessarily have $\sin ( ( n + 1 ) \theta ) = 0$ and $\rho = 1$.

We define the Redheffer matrix $H_n$ and its characteristic polynomial $\chi_n$. We denote $\log_2$ the logarithm function in base 2.
Finally, show that $H_n$ has 1 as an eigenvalue and that its multiplicity is exactly
$$n - \lfloor \log_2 n \rfloor - 1.$$

For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

For $n \in \mathbb { N } ^ { \star }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

We define the Redheffer matrix by $H_n = \left(h_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ where
$$h_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ 1 & \text{if } i \text{ divides } j \text{ and } j \neq 1 \\ 0 & \text{otherwise.} \end{cases}$$
Using the expression
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\left\lfloor \log_2 n \right\rfloor} (\lambda - 1)^{n-k-1} S_k(n),$$
finally show that $H_n$ has 1 as an eigenvalue and that its multiplicity is exactly
$$n - \left\lfloor \log_2 n \right\rfloor - 1.$$

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
Determine the eigenspace of $A_1$ associated with the eigenvalue $\lambda_1$ and deduce the spectrum of $A_1$.

Show that, if $A \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then 0 is an eigenvalue of $A$ and that it is the only complex eigenvalue of $A$.