Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Set $P = QT$ as defined in question II.G.4. Let $Y ^ { \prime } \in \mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $( X , H , Y ^ { \prime } )$ is an admissible triple.
a) Deduce from question II.G.1 the matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X$.
b) Compute the matrices $\Phi _ { X } \left( Y - Y ^ { \prime } \right)$ and $\Phi _ { H } \left( Y - Y ^ { \prime } \right)$.
c) Deduce that we have $Y ^ { \prime } = Y$.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Prove the identity $( X , H , Y ) = \left( P X _ { 0 } P ^ { - 1 } , P H _ { 0 } P ^ { - 1 } , P Y _ { 0 } P ^ { - 1 } \right)$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $f$ be a diagonalizable endomorphism of $V$ and $W$ a non-zero subspace of $V$ stable under $f$. Show that the endomorphism of $W$ induced by $f$ is diagonalizable.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $f$ and $g$ be two endomorphisms of $V$ that commute, that is, such that $f \circ g = g \circ f$. Show that the eigenspaces of $f$ are stable under $g$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $I$ be a non-empty set and let $\left\{ f _ { i } \mid i \in I \right\}$ be a family of diagonalizable endomorphisms of $V$ commuting pairwise. Show that there exists a basis of $V$ in which the matrices of the endomorphisms $f _ { i }$, for $i \in I$, are diagonal. Hint: one may first treat the case where all the endomorphisms $f _ { i }$ are homotheties, then reason by induction on the dimension of $V$.

Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. Let $H$ be an element of $\mathcal { E }$.
a) Calculate the image under $\Phi _ { H }$ of the canonical basis of $\mathcal { M } ( n , \mathbb { K } )$. Deduce that $\Phi _ { H }$ is a diagonalisable endomorphism of $\mathcal { M } ( n , \mathbb { K } )$.
b) Show that there exists a basis of $\mathcal { A }$ in which the matrices of the endomorphisms of $\mathcal { A }$ induced by the $\Phi _ { H }$, for $H \in \mathcal { E }$, are diagonal.

Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. For every map $\lambda$ from $\mathcal { E }$ to $\mathbb { K }$, we set: $$\mathcal { A } _ { \lambda } = \left\{ M \in \mathcal { A } \mid \Phi _ { H } ( M ) = \lambda ( H ) M \text { for all } H \in \mathcal { E } \right\}$$
Let $\lambda$ be a map from $\mathcal { E }$ to $\mathbb { K }$.
a) Show that $\mathcal { A } _ { \lambda }$ is a vector subspace of $\mathcal { A }$.
b) Show that if $\mathcal { A } _ { \lambda }$ is not reduced to $\{ 0 \}$, then $\lambda$ is a linear form on $\mathcal { E }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
Show that $\mathcal { A }$ is a vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket. Show that we have $\mathcal { A } _ { 0 } = \mathcal { E }$, where $\mathcal { A } _ { 0 }$ denotes $\mathcal { A } _ { \lambda }$ when $\lambda$ is the zero linear form. Give a basis of $\mathcal { A } _ { 0 }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
For $k \in \{ 1,2 \}$, we denote by $e _ { k }$ the element of $\mathcal { E } ^ { * }$ which associates to every matrix $\left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right)$, where $D = \left( \begin{array} { c c } d _ { 1 } & 0 \\ 0 & d _ { 2 } \end{array} \right) \in \mathcal { D } ( 2 , \mathbb { R } )$, the coefficient $d _ { k }$.
a) Verify that $(e _ { 1 } , e _ { 2 })$ forms a basis of $\mathcal { E } ^ { * }$.
We equip $\mathcal { E } ^ { * }$ with the unique inner product making $(e _ { 1 } , e _ { 2 })$ an orthonormal basis.
b) Let $\mathcal { R } = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$. Show that the set $\mathcal { R }$ is a root system of $\mathcal { E } ^ { * }$. For this, you may draw the set $\mathcal { R }$ in the Euclidean space $\mathcal { E } ^ { * }$ and recognise one of the root systems encountered in question I.D.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Let $\alpha \in \mathcal { R }$. Determine by calculation the vector subspace $\mathcal { A } _ { \alpha }$. Verify that $\mathcal { A } _ { \alpha }$ is a one-dimensional vector space.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Establish the relation $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Prove the equality $\mathcal { S } ( \mathcal { A } ) = \mathcal { R }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
We now set $\alpha = e _ { 1 } - e _ { 2 }$, $\beta = 2 e _ { 2 }$, $H _ { \alpha } = \left( \begin{array} { c c c c } 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$ and $H _ { \beta } = \left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array} \right)$.
a) Using the results from question III.C.3, show that there exists a pair $\left( X _ { \alpha } , X _ { - \alpha } \right) \in \mathcal { A } _ { \alpha } \times \mathcal { A } _ { - \alpha }$ and a pair $\left( X _ { \beta } , X _ { - \beta } \right) \in \mathcal { A } _ { \beta } \times \mathcal { A } _ { - \beta }$ such that $( X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } )$ and $( X _ { \beta } , H _ { \beta } , X _ { - \beta } )$ are admissible triples of $\mathcal { A }$.
b) Show that $\mathcal { A }$ is the smallest vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket and containing the matrices $X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } , X _ { \beta } , H _ { \beta }$ and $X _ { - \beta }$.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive if and only if all its eigenvalues are positive.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive definite if and only if all its eigenvalues are strictly positive.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.
For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
In the special cases $n = 1$ and $n = 2$, show directly that any matrix $A \in \mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
Let $n \in \mathbb{N}^*$. We assume that any matrix of $\mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite. We consider a matrix $A$ of $\mathcal{S}_{n+1}(\mathbb{R})$ satisfying property $\mathcal{P}_{n+1}$ and we assume by contradiction that $A$ is not positive definite.
a) Show then that $A$ admits two linearly independent eigenvectors associated with eigenvalues (not necessarily distinct) that are strictly negative.
b) Deduce that there exists $X \in \mathcal{M}_{n+1,1}(\mathbb{R})$ whose last component is zero and such that ${}^t X A X < 0$.
c) Conclude.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A$ be a matrix of $\mathcal{S}_n(\mathbb{R})$. Do we have the following equivalence: $$A \text{ is positive} \quad \Longleftrightarrow \quad \forall i \in \llbracket 1; n \rrbracket, \operatorname{det}\left(A^{(i)}\right) \geqslant 0 ?$$

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Calculate $H_2$ and $H_3$. Show that these are invertible matrices and determine their inverses.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Show the relation: $$\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$$
Hint: you may start by subtracting the last column of $\Delta_{n+1}$ from all the others.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Using the relation $\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$, deduce the expression of $\Delta_n$ as a function of $n$ (we will use the quantities $c_m = \prod_{i=1}^{m-1} i!$ for appropriate integers $m$).

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.

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.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm.
Let $n \in \mathbb{N}$. Show that there exists a unique polynomial $\Pi_n \in \mathbb{R}_n[X]$ such that $$\left\|\Pi_n - f\right\| = \min_{Q \in \mathbb{R}_n[X]} \|Q - f\|$$