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

2023 x-ens-maths-b__mp

24 maths questions

Q1 Proof Proof of Stability or Invariance View

Show that, for all $\rho > 0$ and all $m , n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } ) , \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { m , n } ( \mathbb { R } ) \right)$ are closed under addition.

Q2 Proof Proof of Stability or Invariance View

Show that, for all $\rho > 0$ and all $n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ are closed under multiplication.

Q3 Proof Proof That a Map Has a Specific Property View

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$. Show that the map $\mathscr { D } _ { \rho } ( \mathbb { R } ) \rightarrow \mathscr { D } _ { r } ( \mathbb { R } )$ which associates to a function $f$ its restriction to $U _ { r }$ is injective.

Q6 Sequences and Series Properties and Manipulation of Power Series or Formal Series View

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) \neq 0$. The purpose of this question is to show that there exists $r \in \mathbb { R } _ { + } ^ { * } , r \leqslant \rho$ such that $\frac { 1 } { f } \in \mathscr { D } _ { r } ( \mathbb { R } )$.
6a. Show that we can assume without loss of generality that $f ( 0 ) = 1$.
We now write $f ( t ) = \sum _ { i = 0 } ^ { \infty } a _ { i } t ^ { i }$ and we assume that $a _ { 0 } = 1$.
6b. Only in this sub-question, we assume that there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $g \in \mathscr { D } _ { r } ( \mathbb { R } )$ such that $f ( t ) g ( t ) = 1$ for all $t \in U _ { r }$. We write $g ( t ) = \sum _ { i = 0 } ^ { \infty } b _ { i } t ^ { i }$. Show that: $$\left\{ \begin{aligned} & b _ { 0 } = 1 \\ & \text { for } n \geqslant 1 , b _ { n } = - \left( b _ { 0 } a _ { n } + \ldots + b _ { n - 1 } a _ { 1 } \right) \end{aligned} \right.$$
We now define the sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ by the above recurrence formula.
6c. Show that there exists $c \in \mathbb { R } _ { + } ^ { * }$ such that $\left| a _ { n } \right| \leqslant c ^ { n }$ for all $n \in \mathbb { N }$.
6d. Show that $\left| b _ { n } \right| \leqslant ( 2 c ) ^ { n }$ for all $n \in \mathbb { N }$.
6e. Conclude.

Q7 Number Theory Algebraic Structures in Number Theory View

Show that $\mathscr { D } _ { \rho } ( \mathbb { R } )$ is an integral domain.

Q8 Proof Proof That a Map Has a Specific Property View

Let $n , m \in \mathbb { N }$ and let $r , s \in \mathbb { R } _ { + } ^ { * } , r < \rho$.
8a. Show that $\| \cdot \| _ { r , s }$ is a norm on $\mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
8b. Show that if $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { m } [ X ] \right)$, then $P Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n + m } [ X ] \right)$ and $$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$

Q9 Polynomial Division & Manipulation View

Let $A , B \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We assume that $B$ is monic of degree $d \leqslant n$.
9a. Show that there exist elements $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $R \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d - 1 } [ X ] \right)$ uniquely determined such that $A = B Q + R$.
The elements $Q$ and $R$ are called respectively the quotient and the remainder of the Euclidean division of $A$ by $B$.
9b. Let furthermore $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$. Show that, if $\left\| B - X ^ { d } \right\| _ { r , s } < s ^ { d }$, then $$\| Q \| _ { r , s } \leqslant \frac { \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } } \quad \text { and } \quad \| R \| _ { r , s } \leqslant \frac { s ^ { d } \cdot \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } }$$ (One may start by treating the case where $B = X ^ { d }$.)

Q10 Polynomial Division & Manipulation View

We consider $P = f _ { 0 } + f _ { 1 } X + \cdots + f _ { n } X ^ { n } \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$, with $\lambda = 0$, $f _ { 0 } ( 0 ) = \cdots = f _ { d - 1 } ( 0 ) = 0$ and $f _ { d }$ the constant function equal to 1.
Let $F \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d } [ X ] \right)$ be monic and such that $F _ { \mid t = 0 } = X ^ { d }$. Let $R$ be the remainder of the Euclidean division of $P$ by $F$. Show that $F + R$ is monic of degree $d$ and that $( F + R ) _ { \mid t = 0 } = X ^ { d }$.

Q11 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We are furthermore given $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$ and we set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Show that we can choose $r$ and $s$ such that $\alpha _ { 0 } + 2 \varepsilon _ { 0 } \leqslant \frac { 1 } { 3 }$ and $\beta _ { 0 } + \varepsilon _ { 0 } \leqslant \frac { 1 } { 3 }$.

Q12 Sequences and Series Functional Equations and Identities via Series View

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Verify that, for all $i \in \mathbb { N }$, we have the relation: $$\left( 1 - Q _ { i } \right) \cdot R _ { i } = \left( Q _ { i + 1 } - Q _ { i } \right) \cdot F _ { i + 1 } + R _ { i + 1 }$$

Q13 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Show that, for all $i \in \mathbb { N }$, we have $\alpha _ { i + 1 } \leqslant \alpha _ { i } + \varepsilon _ { i }$ and if $\alpha _ { i + 1 } < 1$ then: $$\beta _ { i + 1 } \leqslant \beta _ { i } + \frac { \beta _ { i } \varepsilon _ { i } } { 1 - \alpha _ { i + 1 } } \quad \text { and } \quad \varepsilon _ { i + 1 } \leqslant \frac { \beta _ { i } \varepsilon _ { i } } { 1 - \alpha _ { i + 1 } } .$$

Q14 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

$\alpha _ { i } \leqslant \alpha _ { 0 } + 2 \cdot \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
$\beta _ { i } \leqslant \beta _ { 0 } + \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
$\varepsilon _ { i } \leqslant 2 ^ { - i } \cdot \varepsilon _ { 0 }$.

Q15 Sequences and series, recurrence and convergence Convergence proof and limit determination View

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$.
15a. Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$.
15b. Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.

Q16 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.

Q17 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) > 0$. Show that there exists $\rho _ { f } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { f } \leqslant \rho$ and such that $f > 0$ on $U _ { \rho _ { f } }$ and $\sqrt { f } \in \mathscr { D } _ { \rho _ { f } } ( \mathbb { R } )$.

Q18 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Q19 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.

Q20 Matrices Linear Transformation and Endomorphism Properties View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.
Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:

$\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
$\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.

Q21 Matrices Linear System and Inverse Existence View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20.
Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)

Q22 Matrices Linear Transformation and Endomorphism Properties View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 2 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$. We consider a real number $a \in U _ { \rho _ { 2 } }$.
22a. Show that $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$.
22b. Show the equalities:

$\operatorname { im } \left( B _ { a } U \right) = \operatorname { im } \left( B _ { a } \right) = \operatorname { ker } \left( A _ { a } \right)$ and
$\operatorname { im } \left( A _ { a } V \right) = \operatorname { im } \left( A _ { a } \right) = \operatorname { ker } \left( B _ { a } \right)$.

(One may begin by showing the inclusions from left to right, then use a dimension argument.)

Q23 Matrices Linear Transformation and Endomorphism Properties View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.

Q24 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that, for all $a \in U _ { \rho _ { 2 } }$, the direct sum $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$ of question 22a is orthogonal for the standard inner product on $\mathbb { R } ^ { n }$.

Q25 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ and $Q^{-1} \cdot M \cdot Q = \operatorname{Diag}(M_1, M_2)$.
Show that there exists $\rho _ { 3 } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { 3 } \leqslant \rho _ { 2 }$ and matrices $R _ { 1 } \in \mathrm { GL } _ { d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right) , R _ { 2 } \in \mathrm { GL } _ { n - d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right)$ such that the matrix $Q \cdot \operatorname { Diag } \left( R _ { 1 } , R _ { 2 } \right)$ is orthogonal. (One may use the result of question 17.)

Q26 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

Prove Theorem 2: Let $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and an orthogonal matrix $P \in \mathscr { D } _ { r } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ such that $P ^ { \mathrm { T } } \cdot M \cdot P$ is diagonal.