Show that the application $$\begin{aligned} j : & \mathbb { K } ^ { 3 } \longrightarrow \mathcal { M } _ { 0 } ( 2 , \mathbb { K } ) \\ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) & \longmapsto \left( \begin{array} { c c } x & y + z \\ y - z & - x \end{array} \right) \end{aligned}$$ is an isomorphism of $\mathbb { K }$-vector spaces.

Let $A$ be a non-zero matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. Show that the following properties are equivalent:
i. The matrix $A$ is nilpotent;
ii. The spectrum of $A$ is equal to $\{ 0 \}$;
iii. The matrix $A$ is similar to the matrix $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Does the result that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial remain true for two non-zero matrices of $\mathcal { M } _ { 0 } ( n , \mathbb { C } )$, with $n \geq 3$?

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Let $A$ be a matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$. We assume that its characteristic polynomial equals $X ^ { 2 } + r ^ { 2 }$, where $r$ is a non-zero real number.
a) Justify the existence of a matrix $P \in GL ( 2 , \mathbb { C } )$ satisfying: $ir H _ { 0 } = P ^ { - 1 } A P$. What is the value of the matrix $A ^ { 2 } + r ^ { 2 } I _ { 2 }$?
b) Let $f$ be the endomorphism of $\mathbb { R } ^ { 2 }$ canonically associated with the matrix $A$, that is, which maps a column vector $u$ of $\mathbb { R } ^ { 2 }$ to the vector $A u$. Let $w$ be a non-zero vector of $\mathbb { R } ^ { 2 }$. Prove that the family $\left( \frac { 1 } { r } f ( w ) , w \right)$ is a basis of $\mathbb { R } ^ { 2 }$, and give the matrix of $f$ in this basis.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$ are similar in $\mathcal { M } ( 2 , \mathbb { R } )$ if and only if they have the same characteristic polynomial.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. We equip the vector space $\mathbb { R } ^ { 3 }$ with its canonical Euclidean affine structure and its canonical frame. For every matrix $A$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$, we denote by $\mathcal { Q } _ { A }$ the set of points of $\mathbb { R } ^ { 3 }$ whose image by the application $j$ has the same characteristic polynomial as $A$.
a) Let $r$ be a strictly positive real number. Show that each of the sets $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { r J _ { 0 } }$ and $\mathcal { Q } _ { r H _ { 0 } }$ is a quadric for which an equation will be specified.
b) Draw graphically the appearance of the quadrics $\mathcal { Q } _ { X _ { 0 } } , \mathcal { Q } _ { J _ { 0 } }$ and $\mathcal { Q } _ { H _ { 0 } }$ on the same drawing.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Express the trace of the matrix $M ^ { 2 }$ in terms of the determinant of $M$.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Prove that the matrix $M$ is nilpotent if and only if the trace of the matrix $M ^ { 2 }$ is zero.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. We assume that the matrices $A$ and $[ A , B ]$ commute.
Prove that the matrix $[ A , B ]$ is nilpotent.

Determine the matrices $M$ of $\mathcal { M } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$. What are the matrices $M$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$?
Recall that $X _ { 0 } = \left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$, $H _ { 0 } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)$, $Y _ { 0 } = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right)$, $J _ { 0 } = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$.

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 $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple (i.e., $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$).
Show using questions II.F and II.C that there exists a matrix $Q \in GL ( 2 , \mathbb { K } )$ satisfying $X = Q X _ { 0 } Q ^ { - 1 }$.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Fix a matrix $Q \in GL(2, \mathbb{K})$ satisfying $X = QX_0Q^{-1}$. We define the vectors $u = Q \binom{1}{0}$ and $v = Q \binom{0}{1}$.
a) By computing the vector $[ H , X ] u$ in two different ways, prove that $u$ is an eigenvector of the matrix $H$.
b) By computing the vector $[ H , X ] v$ in two different ways, prove the existence of a scalar $t$ satisfying the identity: $H = Q \left( \begin{array} { c c } 1 & t \\ 0 & - 1 \end{array} \right) Q ^ { - 1 }$.
c) Find a matrix $T \in GL ( 2 , \mathbb { K } )$ commuting with $X _ { 0 }$ and satisfying the relation $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.

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.

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 $A$.
Let $A$ be a matrix of $\mathcal{S}_n(\mathbb{R})$. Do we have the following equivalence: $$A \text{ is positive} \quad \Longleftrightarrow \quad \forall i \in \llbracket 1; n \rrbracket, \operatorname{det}\left(A^{(i)}\right) \geqslant 0 ?$$

Write a procedure, in the Maple or Mathematica language, which takes as input a matrix $M \in \mathcal{S}_n(\mathbb{R})$ and which, using the characterization from I.B, returns ``true'' if the matrix $M$ is positive definite, and ``false'' otherwise.

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: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$
Show that $H_n$ is the matrix of the inner product $\langle \cdot, \cdot \rangle$, restricted to $\mathbb{R}_{n-1}[X]$, in the canonical basis of $\mathbb{R}_{n-1}[X]$.

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 $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Calculate the coefficients of $\Pi_n$ using the matrix $H_{n+1}^{-1}$ and the reals $\langle f, X^i \rangle$.