We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. The hyperplane $\mathcal{H}$ has normal vector $Z$ and equation $x_1 + \cdots + x_n = 0$.
Show that if $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$, then $$\forall X \in \mathcal{H}, \quad {}^t X \Psi(M) X \geqslant \lambda_{\min}\, {}^t XX \quad \text{and} \quad {}^t X \Psi(D) X \geqslant \mu_{\min}\, {}^t XX$$

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$.
Deduce that for $c = \widetilde{c} = -2\mu_{\min} + \sqrt{4\mu_{\min}^2 - 2\lambda_{\min}} > 0$, $\Psi(M_c)$ has non-negative eigenvalues and that for all $c > \widetilde{c}$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We seek the minimal constant $c^* > 0$ (if it exists) satisfying:

• $\Psi(M_{c^*})$ has non-negative eigenvalues,
• for all $c > c^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We know that $c^*$ is bounded above by $\widetilde{c}$.
We consider $\mathcal{A} = \left\{X \in \mathcal{H} \mid \|X\| = 1 \text{ and } 4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X \geqslant 0\right\}$ and we define the mapping $$\alpha: \begin{cases}\mathcal{A} \longrightarrow \mathbb{R} \\ X \longmapsto -2\, {}^t X \Psi(D) X + \sqrt{4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X}\end{cases}$$
Show that there exists $X^* \in \mathcal{A}$ such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X)$ and $\alpha(X^*) > 0$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $X^* \in \mathcal{A}$ be such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X) > 0$, and denote $\alpha^* = \alpha(X^*)$.
Show that:

• ${}^t X^* \Psi(M_{\alpha^*}) X^* = 0$,
• $\Psi(M_{\alpha^*})$ has non-negative eigenvalues,
• for all $c > \alpha^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

Conclude that $c^* = \alpha^*$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $c^* = \alpha^*$ be the minimal constant found previously, and $X^*$ the associated vector.
a) Show that $\Psi(M_{c^*}) X^* = 0$.
We set $Y^* = \frac{2}{c^*} \Psi(M) X^*$.
b) Show that the column vector $\binom{Y^*}{X^*}$ is an eigenvector of the $2n \times 2n$ matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and that $c^*$ is an eigenvalue of this matrix.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We consider $\gamma$ a real eigenvalue of the matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and $\binom{X_1}{X_2}$ an associated eigenvector.
a) Show that ${}^t X_2 \Psi(M_\gamma) X_2 = 0$ and that $X_2 \neq 0$. Conclude that $\gamma \leqslant c^*$.
b) What conclusion do we draw from this on the calculation of the smallest additive constant $c^*$?

Show that there exists a basis $\left( e _ { i } \right) _ { 1 \leq i \leq n }$ of $\mathbb { R } ^ { n }$ and $n$ strictly positive real numbers $\lambda _ { i } \in \mathbb { R } ^ { + * } ( 1 \leq i \leq n )$ such that $$\forall i \in \{ 1 , \ldots , n \} , A ^ { - 1 } K e _ { i } = \lambda _ { i } e _ { i }$$

$\mathcal{P}$ 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.

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that $f_A$ and $f_B$ have the same eigenvalues.

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that if $\lambda$ is an eigenvalue of $A$, then $E_\lambda(f_A) = f_P(E_\lambda(f_B))$.

We denote $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$. Verify that the endomorphism $\sigma_0 = f_{K_2}$ is a reflection (orthogonal symmetry with respect to a line in the plane) and specify its eigenelements.

For all real $t$, specify the endomorphism $\sigma_t$ canonically associated with $R_t^{-1} K_2 R_t$ and in particular its eigenelements.

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 $A \in \mathcal{M}_2(\mathbb{R})$. Compare the eigenvalue circle of $A$ and that of its transpose.

Let $A \in \mathcal{M}_2(\mathbb{R})$. Determine a necessary and sufficient condition on $\mathcal{CP}_A$ for $A$ to be symmetric.

Let $A \in \mathcal{M}_2(\mathbb{R})$.
a) Determine the matrices whose eigenvalue circle has zero radius and characterize geometrically their canonically associated endomorphism.
b) When the eigenvalue circle is reduced to its center, specify the canonically associated endomorphism, on the one hand when this center belongs to the unit circle (with center the origin $O=(0,0)$ and radius 1) and on the other hand when this center belongs to the $x$-axis.
c) What can be said about the matrix $A$ and $f_A$ when the eigenvalue circle $\mathcal{CP}_A$ has zero radius and center belonging to the $y$-axis $\{0\} \times \mathbb{R}$?

Show that two matrices $A$ and $B$ of $\mathcal{M}_2(\mathbb{R})$ are directly orthogonally similar if and only if they have the same eigenvalue circle.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$, draw the circle and the quadrilateral $EHFG$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
When the four points $E, F, G$ and $H$ are distinct show that they are the vertices of a rectangle, which we will call the eigenvalue rectangle of $A$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
Specify the matrices for which some of these points coincide, that is, when the rectangle is flattened.

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. Show that there exists a unique triplet $(\alpha, \beta, \gamma)$ of $\mathbb{R}^2 \times \mathbb{R}_+$ that we will specify, such that $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$.

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 $A$.

Show that for every endomorphism $f$ of $\mathbb{R}^2$, there exist non-negative reals $k$ and $\ell$, a plane rotation $\rho_t$ and a reflection $\sigma_{t'}$ such that $f = k\rho_t + \ell\sigma_{t'}$.

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns a quadruple $(k, \ell, t, t')$ such that if $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ we have $f_A = k\rho_t + \ell\sigma_{t'}$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that $A$ is diagonalizable.