Let $r$ and $s$ be two non-zero natural integers. Let $A \in \mathcal { M } _ { r } ( \mathbb { R } ) , B \in \mathcal { M } _ { r , s } ( \mathbb { R } ) , C \in \mathcal { M } _ { s , r } ( \mathbb { R } )$ and $D \in \mathcal { M } _ { s } ( \mathbb { R } )$. We further assume that $A$ is invertible. We consider the matrix $M \in \mathcal { M } _ { r + s } ( \mathbb { R } )$ having the following block form $$M = \left[ \begin{array} { l l } A & B \\ C & D \end{array} \right]$$ Find two matrices $U \in \mathcal { M } _ { r , s } ( \mathbb { R } )$ and $V \in \mathcal { M } _ { s } ( \mathbb { R } )$ such that $$M = \left[ \begin{array} { c c } A & 0 \\ C & I _ { s } \end{array} \right] \cdot \left[ \begin{array} { c c } I _ { r } & U \\ 0 & V \end{array} \right]$$ and deduce that $$\operatorname { det } ( M ) = \operatorname { det } ( A ) \cdot \operatorname { det } \left( D - C A ^ { - 1 } B \right)$$

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $$a_{1} - a_{2} \geqslant b_{1} - b_{2}.$$ We seek to identify the set $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $\Sigma$ is included in a line segment $L$ of length $\sqrt{2}\left(b_{1} - b_{2}\right)$, and determine its endpoints. One may first study the case where $A$ and $B$ are diagonal.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
For $t \in \mathbb{R}$ and $n \in \mathbb{N}^{*}$, we introduce the matrix $$F_{n}(t) = I_{3} + \sum_{k=1}^{n} \frac{t^{k} \mathcal{M}^{k}}{k!} \in \mathcal{M}_{n}(\mathbb{K})$$
Show that the sequence $\left(F_{n}(t)\right)_{n \in \mathbb{N}}$ converges to a limit denoted $F(t) \in \mathcal{M}_{3}(\mathbb{R})$. Express $F(t)$ in terms of $R(\theta)$, where $\theta$ depends on $m$ and $t$.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $$\Sigma = \left\{s^{\downarrow}(A + B) \,\middle|\, A = \left(\begin{array}{cc} a_{1} & 0 \\ 0 & a_{2} \end{array}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $b_{1} \geqslant b_{2}$.
Determine a continuous function defined on $[-\pi, \pi]$ whose image equals $S\left(b_{1}, b_{2}\right)$.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$ Let $L$ be the line segment identified in question 9.
Show that $\Sigma = L$.

For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$. Show that the set $$\left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\} ,$$ denoted $\mathcal { C } _ { M }$, is a compact subset of $\mathbb { R } ^ { n }$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that for all $t \in \mathbb{R}$ and $X, Y$ vectors of $\mathbb{R}^{3}$, we have $F(t)X \cdot Y = X \cdot F(-t)Y$. Deduce that $F(t)(X \wedge Y) = (F(t)X) \wedge (F(t)Y)$.

For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$, and denote $\mathcal { C } _ { M } = \left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\}$. We further assume that the eigenvalues of $M$ are distinct. We thus have $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$.
(a) Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right)$ such that $\operatorname { Sp } ( M )$ and $\widehat { \mu }$ are strictly interlaced. Show that $\widehat { \mu }$ belongs to $\mathcal { C } _ { M }$.
(b) Show that $$\mathcal { C } _ { M } = \left\{ \widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) , \text { such that } \operatorname { Sp } ( M ) \text { and } \widehat { \mu } \text { are interlaced } \right\} .$$

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M \in S _ { n } ( \mathbb { R } )$. We study the set $$\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\} .$$ We first study the case $n = 2$. We then denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \right)$. Show that $\mathcal { D } _ { M }$ is the line segment in $\mathbb { R } ^ { 2 }$ whose endpoints are $( \lambda _ { 1 } , \lambda _ { 2 } )$ and $( \lambda _ { 2 } , \lambda _ { 1 } )$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Show that for $X \in \mathbb{R}^{3}$, $\mathcal{M}X \cdot X = 0$. Geometrically interpret the application $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3},\, X \mapsto (I_{3} + \mathcal{M})X$.

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M = \left( m _ { i j } \right) \in S _ { n } ( \mathbb { R } )$. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n } \right) \in \mathbb { R } ^ { n }$. We propose to prove that, for all $s \in \{ 1 , \ldots , n \}$, we have: $$\sum _ { i = 1 } ^ { s } m _ { i i } \leqslant \sum _ { i = 1 } ^ { s } \lambda _ { i }$$ (a) What do you think of the case $s = n$ ?
(b) Express $\sum _ { i = 1 } ^ { n - 1 } m _ { i i }$ in terms of the eigenvalues of the matrix $M _ { \leqslant n - 1 }$ obtained by removing the last row and last column of $M$. Deduce inequality (3) when $s = n - 1$.
(c) By proceeding by induction on $n$, show inequality (3), for all $s \in \{ 1 , \ldots , n \}$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Give a necessary and sufficient condition on $m$ for the sequence $\left((I_{3} + \mathcal{M})^{n}\right)_{n \in \mathbb{N}}$ to converge in $\mathcal{M}_{3}(\mathbb{R})$.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$. Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$.
(a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$.
(b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$.
(c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12. Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$.
(d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$.
(e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.

Prove that for all $M \in \mathcal{Y}_n$, $\operatorname{det}(M) \leqslant n!$ and that there is no equality.

Prove that $\mathcal{Y}_n$ is a convex and compact subset of $\mathcal{M}_n(\mathbb{R})$.

Let $M \in \mathcal{Y}_n$ and $\lambda$ a complex eigenvalue of $M$. Prove that $|\lambda| \leqslant n$ and give an explicit example where equality holds.

List the elements of $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$. Specify (by justifying) which ones are diagonalizable over $\mathbb{R}$.

Prove that $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$ generates the vector space $\mathcal{M}_2$. For $n \geqslant 2$, does $\mathcal{X}_n'$ generate the vector space $\mathcal{M}_n(\mathbb{R})$?

For all $(M, N) \in (\mathcal{M}_n(\mathbb{R}))^2$, we denote $$(M \mid N) = \operatorname{tr}({}^t M N)$$ Prove that this defines an inner product on $\mathcal{M}_n(\mathbb{R})$. Explicitly express $(M \mid N)$ in terms of the coefficients of $M$ and $N$.

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Fix $A \in \mathcal{M}_n(\mathbb{R})$, prove that there exists a matrix $M \in \mathcal{Y}_n$ such that: $$\forall N \in \mathcal{Y}_n \quad \|A - M\| \leqslant \|A - N\|$$

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Justify the uniqueness of the matrix $M \in \mathcal{Y}_n$ minimizing $\|A - M\|$ over $\mathcal{Y}_n$ and explicitly express its coefficients in terms of those of $A$.

Justify that the determinant has a maximum on $\mathcal{X}_n$ (denoted $x_n$) and a maximum on $\mathcal{Y}_n$ (denoted $y_n$).

Let $J \in \mathcal{X}_n$ be the matrix whose coefficients all equal 1. We set $M = J - I_n$.
Calculate $\operatorname{det}(M)$ and deduce that $\lim_{k \to +\infty} y_k = +\infty$.

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.
Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.
Deduce that $x_n = y_n$.