grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2014 centrale-maths2__psi

27 maths questions

QI.A.1 Matrices Matrix Group and Subgroup Structure View
Does the matrix $\Delta _ { p + 1 }$ belong to the set $O ( 1 , p )$ ? to the set $O ^ { + } ( 1 , p )$ ?
QI.A.2 Groups Subgroup and Normal Subgroup Properties View
Show that $O ( 1 , p ) = O ^ { + } ( 1 , p ) \cup O ^ { - } ( 1 , p )$.
QI.A.3 Groups Subgroup and Normal Subgroup Properties View
Show that the set $O ( 1 , p )$ is a subgroup of $G L _ { p + 1 } ( \mathbb { R } )$ and that $O ^ { + } ( 1 , p )$ is a subgroup of $O ( 1 , p )$.
QI.A.4 Matrices Matrix Group and Subgroup Structure View
Show that, for every matrix $L$ element of $O ( 1 , p )$, its transpose ${ } ^ { t } L$ is also an element of $O ( 1 , p )$.
QI.A.5 Matrices Matrix Group and Subgroup Structure View
Show that the sets $O ( 1 , p ) , O ^ { + } ( 1 , p )$ and $O ^ { - } ( 1 , p )$ of $\mathcal { M } _ { p + 1 } ( \mathbb { R } )$ are closed.
QI.B.1 Matrices Matrix Algebra and Product Properties View
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 $A = B$.
QI.B.2 Matrices Bilinear and Symplectic Form Properties View
We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Express $\varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$ as a function of $q _ { p + 1 } \left( v + v ^ { \prime } \right)$ and $q _ { p + 1 } \left( v - v ^ { \prime } \right)$.
QI.B.3 Groups Symplectic and Orthogonal Group Properties View
We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated.
Show that the following three assertions are equivalent:
i. $L \in O ( 1 , p )$;
ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$;
iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.
QI.B.4 Groups Symplectic and Orthogonal Group Properties View
We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to $$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$ What do we obtain similarly with ${ } ^ { t } L$ ?
QII.A.1 Hyperbolic functions View
Let $a$ and $b$ be two real numbers. If $a > 0$ and $a ^ { 2 } - b ^ { 2 } = 1$ show that there exists a unique $\theta \in \mathbb { R }$ such that $a = \operatorname { ch } \theta$ and $b = \operatorname { sh } \theta$.
QII.A.2 Groups Symplectic and Orthogonal Group Properties View
Let $a , b , c$ and $d$ be four real numbers. We consider the matrix of $\mathcal { M } _ { 2 } ( \mathbb { R } )$ $$L = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$$ Write the equations on $a , b , c , d$ expressing the membership of $L$ in $O ( 1,1 )$.
QII.A.3 Hyperbolic functions View
Deduce the equality: $$O ^ { + } ( 1,1 ) = \left\{ \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right) , \gamma \in \mathbb { R } \right\} \cup \left\{ \left( \begin{array} { c c } - \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & - \operatorname { ch } \gamma \end{array} \right) , \gamma \in \mathbb { R } \right\}$$
QII.A.4 Groups Subgroup and Normal Subgroup Properties View
We denote, for every real $\gamma$, $L ( \gamma ) = \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right)$.
Show, for all real numbers $\gamma$ and $\gamma ^ { \prime }$, the equality: $$L ( \gamma ) L \left( \gamma ^ { \prime } \right) = L \left( \gamma + \gamma ^ { \prime } \right)$$ Deduce that $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ is a commutative subgroup of the group $O ^ { + } ( 1,1 )$.
QII.B Groups Group Order and Structure Theorems View
Is the group $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ compact?
QII.C Matrices Diagonalizability and Similarity View
Show that the matrices that are elements of $O ^ { + } ( 1,1 )$ are diagonalizable and find a matrix $P \in O ( 2 )$ such that, for every matrix $L \in O ^ { + } ( 1,1 )$, the matrix ${ } ^ { t } P L P$ is diagonal.
QII.D Groups Group Order and Structure Theorems View
Show that the group $O ^ { + } ( 1,1 )$ is commutative.
QIII.A Groups Symplectic and Orthogonal Group Properties View
Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ( 1,3 )$. Show the inequality $\ell _ { 1,1 } ^ { 2 } \geqslant 1$.
QIII.B Groups Subgroup and Normal Subgroup Properties View
Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 }$ and $L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$ be two elements of $\tilde { O } ( 1,3 )$. We set $L ^ { \prime \prime } = L L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$.
Prove the following inequalities: $$0 \leqslant \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } ^ { 2 } } \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { k , 1 } ^ { \prime 2 } } + \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } \ell _ { k , 1 } ^ { \prime } < \ell _ { 1,1 } ^ { \prime \prime }$$ Deduce that the set $\tilde { O } ( 1,3 )$ is a subgroup of the Lorentz group $O ( 1,3 )$.
QIII.C Groups Group Homomorphisms and Isomorphisms View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Justify that $G$ is a subgroup of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ isomorphic to $S O ( 3 )$.
QIII.D Groups Subgroup and Normal Subgroup Properties View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$.
Show that, if the vector $a$ is zero, then the matrix $L$ belongs to the group $G$.
QIII.E.1 Groups Symplectic and Orthogonal Group Properties View
In the usual Euclidean space $\mathbb { R } ^ { 3 }$, show that, for all vectors $u$ and $v$ of $\mathbb { R } ^ { 3 }$ of the same norm, there exists a rotation $r$ such that $r ( u ) = v$.
QIII.F.1 Groups Group Homomorphisms and Isomorphisms View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero.
Deduce from question III.E.1 that there exists an element $L _ { 1 }$ of $G$ such that: $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ where $\alpha$ is a strictly positive real number that we will specify, $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ are real numbers that we will not seek to determine.
QIII.F.2 Groups Group Homomorphisms and Isomorphisms View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $v _ { 2 } = \left( \begin{array} { l } \mu _ { 1 } \\ \mu _ { 2 } \\ \mu _ { 3 } \end{array} \right)$ and $v _ { 3 } = \left( \begin{array} { l } \nu _ { 1 } \\ \nu _ { 2 } \\ \nu _ { 3 } \end{array} \right)$. Show that $v _ { 2 }$ and $v _ { 3 }$ are two unit vectors orthogonal to each other in $\mathbb { R } ^ { 3 }$ equipped with its usual Euclidean structure.
QIII.F.3 Groups Group Homomorphisms and Isomorphisms View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $R _ { 2 } \in S O ( 3 )$. We set $L _ { 2 } = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right) \in G$. Show that we can choose $R _ { 2 }$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \delta _ { 1 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are real numbers that we will not seek to determine.
QIII.F.4 Groups Subgroup and Normal Subgroup Properties View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix $L_1 \in G$ and $R_2 \in SO(3)$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $L_2 = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right)$.
Show that the real numbers $\beta _ { 2 } , \beta _ { 3 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are zero.
QIII.G Groups Group Order and Structure Theorems View
We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Deduce that every matrix $L$ of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ can be written in the form of a product of the type $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ where $R$ and $R ^ { \prime }$ are two elements of $S O ( 3 )$ and $\gamma$ is a real number.
QIII.I Groups True/False with Justification View
Is the decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ obtained unique?