For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 5 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of matrix $A - B$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two matrices $A = \left( \begin{array} { l l } a & 3 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { r r } 4 & 1 \\ - 1 & 0 \end{array} \right)$, when the sum of all entries of matrix $A + B$ is 9, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $P$ be a matrix of $GL ( 2 , \mathbb { K } )$. Verify that $( P X _ { 0 } P ^ { - 1 } , P H _ { 0 } P ^ { - 1 } , P Y _ { 0 } P ^ { - 1 } )$ is an admissible triple.
Recall that a triple $(X, H, Y)$ of three non-zero matrices of $\mathcal{M}(n, \mathbb{K})$ is an admissible triple if $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Let $Q$ be a real orthogonal matrix. Prove that $R ( A ) = R \left( { } ^ { t } Q A Q \right)$.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
Justify that ${}^t A = U({}^t B)U^{-1}$.

For every $A \in \mathcal{M}_d(\mathbb{R})$, we define $$\cos(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n}}{(2n)!} \quad \text{and} \quad \sin(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n+1}}{(2n+1)!}$$ Show $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad \cos(A)^2 + \sin(A)^2 = I_d$$

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = \alpha_{i,k,j} = -\alpha_{k,j,i}$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Deduce that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = 0$.

We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$. We further assume that $A$ and $B$ commute with $[A,B]$.
(a) Show that $[A, \exp(B)] = \exp(B)[A,B]$.
(b) Determine a differential equation satisfied by $t \mapsto \exp(tA)\exp(tB)$.
(c) Deduce the formula: $$\exp(A)\exp(B) = \exp\left(A + B + \frac{1}{2}[A,B]\right)$$

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. a) Calculate the products pairwise of the matrices $e, I, J, K$. Present the results in a double-entry table. b) Deduce that $(\mathrm{i} I, \mathrm{i} J, \mathrm{i} K)$ is an H-system.

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Verify that, for all $q \in \mathbb{H}$, $q^*$ is the transpose of the matrix whose coefficients are the conjugates of the coefficients of $q$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $(qr)^* = r^* q^*$. c) Show that $q^{**} = q$ for all $q \in \mathbb{H}$ and that $q \mapsto q^*$ is an automorphism of the $\mathbb{R}$-vector space $\mathbb{H}$. d) For $q \in \mathbb{H}$, express $qq^*$ in terms of $N(q)$. Deduce the relation valid for all $(q, r) \in \mathbb{H}^2$ $$N(qr) = N(q)N(r)$$

For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Let $(x, y, z, t) \in \mathbb{R}^4$ and $q = xe + yI + zJ + tK$. Express the trace of the matrix $q \in \mathcal{M}_2(\mathbb{C})$ in terms of the real number $x$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $qr - rq = q^* r^* - r^* q^*$. c) Let $a, b, c, d$ be quaternions. Establish the relation $(acb^*)d + d^*(acb^*)^* = (acb^*)^* d^* + d(acb^*)$.
Deduce the identity $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$. Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.
You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.

Let $A$ and $B$ be two matrices of $\mathcal { M } _ { n } ( \mathbb { R } )$. Show that if, for all $X$ and $Y$ of $\mathbb { R } ^ { n } , { } ^ { t } X A Y = { } ^ { t } X B Y$ then $A = B$.

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 $F(t)(X \wedge Y) = (F(t)X) \wedge (F(t)Y)$.

Let $(\Omega, \mathscr{A}, \mathbf{P})$ be a probability space and $X : \Omega \rightarrow \{1, \ldots, N\}$ a random variable with distribution $q \in \Sigma_{N}$. Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$ for $(i,j) \in \{1, \ldots, N\} \times \{1, \ldots, d\}$, $p \in \Sigma_{N}$ and $m \in \mathbb{R}^{d}$. We denote by $A \in \mathscr{M}_{d}(\mathbb{R})$ the square matrix of size $d \times d$ defined for all $(k,l) \in \{1, \ldots, d\}^{2}$ by $$A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k}).$$
Verify that if $Y : \Omega \rightarrow \{1, \ldots, N\}$ is a random variable with distribution $p$, then $A_{lk} = \mathbf{E}((g_{l}(Y) - m_{l})(g_{k}(Y) - m_{k}))$ and then that $A$ is a symmetric matrix such that $\theta^{T} A \theta \geqslant 0$ for all $\theta \in \mathbb{R}^{d}$.

Show that $\left\| z _ { 1 } - z _ { 2 } \right\| = \left\| z _ { 1 } + z _ { 2 } \right\|$ and that $- \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } = 0$.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that ${ } ^ { t } P P + q { } ^ { t } q = I _ { n - 1 }$, ${ } ^ { t } P r + \alpha q = 0$ and ${ } ^ { t } r r + \alpha ^ { 2 } = 1$.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that $${ } ^ { t } Q _ { + } Q _ { - } = I _ { n - 1 } - \frac { 2 } { 1 - \alpha ^ { 2 } } q { } ^ { t } q$$ and deduce that $$Q _ { - } = Q _ { + } - \frac { 2 } { 1 - \alpha ^ { 2 } } Q _ { + } q { } ^ { t } q$$

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ Using the induction hypothesis for $Q _ { + }$ (resp. for $Q _ { - }$), we denote by $x _ { + } > 0$ (resp. $x _ { - } > 0$) a vector of $\mathbb { R } ^ { n - 1 }$ and $S _ { + }$ (resp. $S _ { - }$) the sign diagonal matrix, such that $$Q _ { + } x _ { + } = S _ { + } x _ { + } , \quad \text { resp. } \quad Q _ { - } x _ { - } = S _ { - } x _ { - }$$
Show that $$\left( S _ { + } x _ { + } \mid S _ { - } x _ { - } \right) = \left( x _ { + } \mid x _ { - } \right) - \frac { 2 } { 1 - \alpha ^ { 2 } } \left( x _ { + } \mid q \right) \left( x _ { - } \mid q \right)$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we propose to show that there does not exist a linear application $u : \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X]$ such that $u \circ u = \delta$. We suppose, by contradiction, that such an application $u$ exists.
a) Show that $u$ and $\delta^2$ commute.
b) Deduce that $\mathbb{R}_1[X]$ is stable under the application $u$.
c) Show that there does not exist a matrix $A \in \mathcal{M}_2(\mathbb{R})$ such that $$A^2 = \left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$
d) Conclude.

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,1}(\mathbb{R}), B_{3} \in \mathcal{M}_{1,n}(\mathbb{R})$, $B_{4} \in \mathcal{M}_{1}(\mathbb{R})$ such that: $A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,2}(\mathbb{R}), B_{3} \in \mathcal{M}_{2,n}(\mathbb{R})$ and $B_{4} \in \mathcal{M}_{2}(\mathbb{R})$ such that $$A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$$

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $\operatorname{det}\left(N^{\top} A^{-1} N\right) > 0$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.