Let $W = \left\{ ( a , b , c , d ) \in \mathbb { R } ^ { 4 } \mid 3 a - b + 6 c = 0 \right\}$ and $T : \mathbb { R } ^ { 4 } \longrightarrow W$ be a linear map with $T ^ { 2 } = T$. Suppose $T$ is onto. Pick the correct statement(s) from below.
(A) $T ( u + v ) = T ( u ) + v$ for all $u \in \mathbb { R } ^ { 4 } , v \in W$.
(B) $\operatorname { ker } ( T - I )$ contains three linearly independent vectors.
(C) $( 1,3,0,2 ) \in \operatorname { ker } ( T )$.
(D) If $v _ { 1 } , v _ { 2 } \in \operatorname { ker } ( T )$ are nonzero, then $v _ { 1 } = c v _ { 2 }$ for some $c \in \mathbb { R }$.

For $x , y \in \mathbb { R } ^ { n }$, we set: $( x ; y ) _ { A } = \langle A x ; y \rangle$. We denote by $E$ the endomorphism of the vector space $\mathbb { R } ^ { n }$ defined by $\forall x \in \mathbb { R } ^ { n } , E ( x ) = A ^ { - 1 } K x$.
Prove that $( ; ) _ { A }$ defines an inner product on $\mathbb { R } ^ { n }$. Then show that $$\forall x , y \in \mathbb { R } ^ { n } , ( E ( x ) ; y ) _ { A } = ( x ; E ( y ) ) _ { A } .$$

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
We consider the following conditions:
(C1) $\quad \operatorname { Tr } ( A ) \in R ( A )$
(C2) There exists a real orthogonal matrix $Q$ such that the diagonal of the matrix ${ } ^ { t } Q A Q$ is of the form $( \operatorname { Tr } ( A ) , 0 , \ldots , 0 )$
I.F.1) Prove that condition (C2) implies condition (C1).
I.F.2) We assume that $x \in R ( A )$.
Prove that there exists an orthogonal matrix $Q _ { 1 }$ such that $${ } ^ { t } Q _ { 1 } A Q _ { 1 } = \left( \begin{array} { c c } x & L \\ C & B \end{array} \right)$$ where $B$ is a matrix of format $( n - 1 , n - 1 )$ $\left( B \in \mathcal { M } _ { n - 1 } ( \mathbb { R } ) \right)$, $C$ a column vector with $n - 1$ elements $\left( C \in \mathcal { M } _ { n - 1,1 } ( \mathbb { R } ) \right)$ and $L$ a row vector with $n - 1$ elements $\left( L \in \mathcal { M } _ { 1 , n - 1 } ( \mathbb { R } ) \right)$.
I.F.3) Prove that if the matrix $A$ is symmetric then so is the matrix $B$ above.
I.F.4) Prove that $\operatorname { Tr } ( A ) = \operatorname { Tr } \left( { } ^ { t } Q _ { 1 } A Q _ { 1 } \right)$.
I.F.5) Deduce that if $A$ is symmetric, condition (C1) implies condition (C2).
One may reason by induction on $n$.

Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities, and $D$ the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.
Deduce that $M_n = \sum_{k=1}^n \sqrt{\mu_k}$.

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
V.A.1) Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal.
This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$.
V.A.2) If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$.
If $\omega \in \Omega$, we introduce the column matrix $$U(\omega) = \begin{pmatrix} X_1(\omega) \\ \vdots \\ X_n(\omega) \end{pmatrix}$$ and the matrix $M(\omega) = U(\omega)\, {}^t(U(\omega))$. The application $M : \left\{\begin{array}{l} \Omega \rightarrow \mathcal{M}_n(\mathbb{R}) \\ \omega \mapsto M(\omega) \end{array}\right.$ is thus a random variable.
a) If $\omega \in \Omega$, justify that $M(\omega) \in \mathcal{X}_n$.
b) If $\omega \in \Omega$, justify that $\operatorname{tr}(M(\omega)) \in \{0, \ldots, n\}$, that $M(\omega)$ is diagonalizable over $\mathbb{R}$ and that $\operatorname{rg}(M(\omega)) \leqslant 1$.
c) If $\omega \in \Omega$, justify that $M(\omega)$ is an orthogonal projection matrix if and only if $S(\omega) \in \{0,1\}$.

Show that $\mathcal{S}_{n}(\mathbb{R})$ and $\mathcal{A}_{n}(\mathbb{R})$ are two supplementary orthogonal vector subspaces in $\mathcal{M}_{n}(\mathbb{R})$ and specify their dimensions. (The inner product on $\mathcal{M}_{n}(\mathbb{R})$ is given by $(M,N) \mapsto \operatorname{tr}(M^{\top}N)$.)

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R})$, $\|A - A_{s}\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.

If $M \in \mathcal{M}_{n}(\mathbb{R})$ and $X, Y \in \mathcal{M}_{n,1}(\mathbb{R})$, the matrix $X^{\top}MY$ belongs to $\mathcal{M}_{1}(\mathbb{R})$ and we agree to identify it with the real number equal to its unique entry.
With this convention, show that $A_{s} \in \mathcal{S}_{n}^{+}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}), X^{\top}A_{s}X \geqslant 0$ and that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}) \setminus \{0\}, X^{\top}A_{s}X > 0$.

Show that $\mathcal{S}_{n}(\mathbb{R})$ and $\mathcal{A}_{n}(\mathbb{R})$ are two supplementary orthogonal vector subspaces in $\mathcal{M}_{n}(\mathbb{R})$ and specify their dimensions.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R}), \left\|A - A_{s}\right\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.

If $M \in \mathcal{M}_{n}(\mathbb{R})$ and $X, Y \in \mathcal{M}_{n,1}(\mathbb{R})$, the matrix $X^{\top} M Y$ belongs to $\mathcal{M}_{1}(\mathbb{R})$ and we agree to identify it with the real number equal to its unique entry.
With this convention, show that $A_{s} \in \mathcal{S}_{n}^{+}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}), X^{\top} A_{s} X \geqslant 0$ and that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}) \backslash \{0\}, X^{\top} A_{s} X > 0$.

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 $\mathcal{M}_{n}(\mathbb{R})$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that $A$ is $H$-regular for every hyperplane $H$ of $E_{n}$.

We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Determine a hyperplane $H$ such that $A(1)$ is $H$-singular.

We return to the example of subsection II.B with $\mu = 1$, i.e., $$A(1) = \left(\begin{array}{ccc} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)$$ How should we choose $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ so that $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = 0$?

Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that if $X \in \mathcal{M}_{p,1}(\mathbb{R})$ is non-zero then $X^{\top} N^{\prime\top} A N^{\prime} X > 0$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. Let $G$ be the set of elements $u \in E$ satisfying: (a) $u \in \operatorname{Im}(T)$, (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Deduce that the map $(v,w) \in G \times G \mapsto (T(v) \mid T(w))$ is a scalar product on $G$.

We equip $\mathcal{M}_n(\mathbb{R})$ with its usual inner product defined by: $\forall (M_1, M_2) \in \mathcal{M}_n(\mathbb{R}), \langle M_1, M_2 \rangle = \operatorname{tr}({}^t M_1 M_2)$. We denote by $\mathcal{S}_{k+1}$ the restriction of $\mathcal{S}$ to $\Delta_{k+1}$ and $\mathcal{S}_k^*$ the restriction of $\mathcal{S}^*$ to $\Delta_k$.
Verify that for all $X$ in $\Delta_{k+1}$ and $Y$ in $\Delta_k$, $\langle \mathcal{S}_{k+1} X, Y \rangle = \langle X, \mathcal{S}_k^* Y \rangle$. Deduce that $\ker(\mathcal{S}_k^*)$ and $\operatorname{Im}(\mathcal{S}_{k+1})$ are orthogonal complements in $\Delta_k$, that is $$\Delta_k = \ker(\mathcal{S}_k^*) \oplus^{\perp} \operatorname{Im}(\mathcal{S}_{k+1})$$

Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, $P_{V,W}$ denotes the matrix of the projection onto $\operatorname{Im}(V)$ parallel to $\operatorname{Im}(W)^\perp$ (when $W^T V$ is invertible).
Show that there exists a neighborhood $\mathscr { V }$ of $V$ in $\mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $W ^ { \mathrm { T } } V$ is invertible for all $W \in \mathscr { V }$ and the map $W \mapsto P _ { V , W }$ is continuous from $\mathscr { V }$ to $\mathscr { M } _ { p } ( \mathbb { R } )$.

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$ and $\pi_A$ the orthogonal projection onto $T_A$.
Let $\phi : \mathscr { M } _ { n , p } ( \mathbb { R } ) \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } ) \times \mathscr { M } _ { p , n } ( \mathbb { R } )$ defined by $\phi ( \tilde { A } ) = \left( \tilde { A } V V ^ { \mathrm { T } } , \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } } \right)$ for all $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$.
(a) Identify $\operatorname { ker } ( \phi )$ in terms of $N _ { A }$.
(b) We denote by $\pi _ { A } : \mathscr { M } _ { n , p } ( \mathbb { R } ) \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } )$ the orthogonal projection onto $T _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$. Show that $\phi = \phi \circ \pi _ { A }$.
(c) Let $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ satisfying condition (C). We denote by $W = \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } }$. Show that if $P _ { V , W }$ is the matrix of the projection onto $\operatorname { Im } ( V )$ parallel to $\operatorname { Im } ( W ) ^ { \perp }$ then $$\tilde { A } = \tilde { A } V V ^ { \mathrm { T } } P _ { V , W }$$

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $\pi_A$ be the orthogonal projection onto $T_A$ in $\mathscr{M}_{n,p}(\mathbb{R})$.
Deduce that there exists $\epsilon > 0$ such that the restriction of $\pi _ { A }$ to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } ) \cap B ( A , \epsilon )$ is injective where $B ( A , \epsilon ) = \left\{ \tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \| \tilde { A } - A \| _ { F } < \epsilon \right\}$ is the open ball of $\mathscr { M } _ { n , p } ( \mathbb { R } )$ centered at $A$ with radius $\epsilon$.

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$.
Let $\rho _ { A }$ be the orthogonal projection onto $N _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$.
(a) Show that for all $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$, we have $\rho _ { A } ( \tilde { A } ) = \left( I _ { n } - U U ^ { \mathrm { T } } \right) \tilde { A } \left( I _ { p } - V V ^ { \mathrm { T } } \right)$.
(b) Show that $\rho _ { A } ( A B ) = 0$ for all $B \in \mathscr { M } _ { p } ( \mathbb { R } )$.
Let $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ satisfy condition (C).
(c) Show that if $W = \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } }$ $$\rho _ { A } ( \tilde { A } ) = \left( I _ { n } - U U ^ { \mathrm { T } } \right) ( \tilde { A } - A ) V V ^ { \mathrm { T } } \left( P _ { V , W } - P _ { V , V } \right) \left( I _ { p } - V V ^ { \mathrm { T } } \right) .$$
(d) Deduce that $\left\| \rho _ { A } ( \tilde { A } ) \right\| _ { F } \leqslant \sqrt { ( n - k ) k ( p - k ) } \| \tilde { A } - A \| _ { F } \left\| P _ { V , W } - P _ { V , V } \right\| _ { F }$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Let $f' \in \mathrm{O}(E)$ having the same characteristic polynomial as $f$. Show that there exist orthonormal bases $\mathcal{B}$ and $\mathcal{B}'$ of $E$ for which the matrix of $f$ in $\mathcal{B}$ is equal to the matrix of $f'$ in $\mathcal{B}'$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Let $f' \in \mathrm{O}(E)$ having the same characteristic polynomial as $f$. Show that there exist orthonormal bases $\mathcal{B}$ and $\mathcal{B}'$ of $E$ for which the matrix of $f$ in $\mathcal{B}$ is equal to the matrix of $f'$ in $\mathcal{B}'$.