grandes-ecoles

QIA Groups Symplectic and Orthogonal Group Properties View

Let $\alpha$ be a non-zero element of $E$. Show, for every vector $x$ of $E$, the identity: $$\tau _ { \alpha } ( x ) = x - 2 \frac { \langle \alpha , x \rangle } { \langle \alpha , \alpha \rangle } \alpha$$

QIB Groups Subgroup and Normal Subgroup Properties View

We assume in this question that the space $E$ has dimension 1. Show that the root systems of $E$ are the sets $\{ \alpha , - \alpha \}$, with $\alpha \in E \backslash \{ 0 \}$.

QIC1 Groups Symplectic and Orthogonal Group Properties View

In this question, the space $E$ has dimension $n \geq 2$. For every pair $(\alpha , \beta)$ of non-zero vectors of $E$, let $\theta _ { \alpha , \beta }$ be the geometric angle between $\alpha$ and $\beta$, that is, the unique element of $[ 0 , \pi ]$ given by: $\| \alpha \| . \| \beta \| \cos \theta _ { \alpha , \beta } = \langle \alpha , \beta \rangle$.
Let $\mathcal { R }$ be a root system of $E$ and let $\alpha , \beta$ be two non-collinear elements of $\mathcal { R }$.
a) Show, using property 4, that: $2 \frac { \| \alpha \| } { \| \beta \| } \left| \cos \theta _ { \alpha , \beta } \right| .2 \frac { \| \beta \| } { \| \alpha \| } \left| \cos \theta _ { \alpha , \beta } \right| \leq 3$.
b) Assume $\| \alpha \| \leq \| \beta \|$. Show that the pair $(\alpha , \beta)$ is found in one of the configurations listed in the table below (each row corresponding to a configuration):

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\\| \beta \\| / \\| \alpha \\|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

QIC2 Groups Symplectic and Orthogonal Group Properties View

In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\\| \beta \\| / \\| \alpha \\|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

QID1 Groups Group Order and Structure Theorems View

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

QID2 Groups Group Order and Structure Theorems View

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ For each value of $k \in \{ 2,3,4,6 \}$, draw graphically a root system $\mathcal { R } _ { k }$ such that $\theta _ { \mathcal { R } _ { k } } = \pi / k$. It is not necessary to justify that the figures drawn represent root systems. What is the cardinality of $\mathcal { R } _ { k }$? No justification is required.

QIE1 Groups Subgroup and Normal Subgroup Properties View

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Show that the vector subspace of $E$ spanned by the set $\mathcal { R } _ { 0 }$ is a vector plane.

QIE2 Groups Group Homomorphisms and Isomorphisms View

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Draw graphically $\mathcal { R } _ { 0 }$ in the plane $\operatorname { Vect } \left( \mathcal { R } _ { 0 } \right)$. Recognize one of the root systems represented in question I.D.2.

QIIA1 Matrices Linear Transformation and Endomorphism Properties View

Show that $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$ is a $\mathbb { K }$-vector space; specify its dimension.

QIIA2 Matrices Determinant and Rank Computation View

Justify that, for every pair $(A , B)$ of elements of $\mathcal { M } ( n , \mathbb { K } )$, the matrix $[ A , B ]$ belongs to $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$.

QIIB Matrices Linear Transformation and Endomorphism Properties View

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.

QIIC Matrices Diagonalizability and Similarity View

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 $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

QIID1 Matrices Diagonalizability and Similarity View

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.

QIID2 Matrices Diagonalizability and Similarity View

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 $n \geq 3$?

QIIE1 Matrices Diagonalizability and Similarity View

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.

QIIE2 Matrices Diagonalizability and Similarity View

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.

QIIE3 Matrices Eigenvalue and Characteristic Polynomial Analysis View

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.

QIIF1 Matrices Determinant and Rank Computation View

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 $M$.

QIIF2 Matrices Linear Transformation and Endomorphism Properties View

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.

QIIF3 Matrices Linear Transformation and Endomorphism Properties View

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.

QIIG1 Matrices Structured Matrix Characterization View

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 $X _ { 0 } = \left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$, $H _ { 0 } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)$, $Y _ { 0 } = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right)$, $J _ { 0 } = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$.

QIIG2 Matrices Matrix Algebra and Product Properties View

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 $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$.

QIIG3 Matrices Diagonalizability and Similarity View

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 $X = Q X _ { 0 } Q ^ { - 1 }$.

QIIG4 Matrices Eigenvalue and Characteristic Polynomial Analysis View

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 $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.

QIIG5 Matrices Linear Transformation and Endomorphism Properties View

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 $Y ^ { \prime } = Y$.

QIIG6 Matrices Diagonalizability and Similarity View

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 $( X , H , Y ) = \left( P X _ { 0 } P ^ { - 1 } , P H _ { 0 } P ^ { - 1 } , P Y _ { 0 } P ^ { - 1 } \right)$.

QIIIA1 Matrices Diagonalizability and Similarity View

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.

QIIIA2 Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis View

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 $g$.

QIIIA3 Invariant lines and eigenvalues and vectors Simultaneous diagonalization or commutant structure View

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 $V$.

QIIIB1 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

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.

QIIIB2 Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis View

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 $\mathcal { E }$.

QIIIC1 Groups Algebra and Subalgebra Proofs View

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 $\mathcal { A } _ { 0 }$.

QIIIC2 Groups Decomposition and Basis Construction View

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.

QIIIC3 Groups Decomposition and Basis Construction View

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.

QIIIC4 Groups Decomposition and Basis Construction View

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 $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.

QIIIC5 Groups Subgroup and Normal Subgroup Properties View

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 $\mathcal { S } ( \mathcal { A } ) = \mathcal { R }$.

QIIIC6 Groups Subgroup and Normal Subgroup Properties View

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 $X _ { - \beta }$.

grandes-ecoles

Papers (191)

2010 centrale-maths2__pc