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
2013 x-ens-maths__pc

22 maths questions

Q1 Linear transformations View
Suppose in this question that $n = 3$. What geometric interpretation can be given to $R_{p,q}(\theta)$?
Q2 Matrices Matrix Group and Subgroup Structure View
Calculate the product ${}^t\left(R_{p,q}(\theta)\right) R_{p,q}(\theta)$. What property of $R_{p,q}(\theta)$ is recognized?
Q3 Matrices Diagonalizability and Similarity View
We are given $S \in \mathbf{S}_n$ and $R \in \mathbf{O}_n$. Verify that ${}^t R S R$ is symmetric and that it is similar to $S$.
Q4 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in \mathbf{M}_n$ and $U, V \in \mathbf{O}_n$. Show that $\|UAV\| = \|A\|$.
Q5 Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
We are given four real numbers $a \leqslant b \leqslant c \leqslant d$ such that $a + d = b + c$. Study the variations of the function $x \mapsto |x - a| - |x - b| - |x - c| + |x - d|$; show that it takes positive values. A reasoned argument supported by a graphical representation would be welcome.
Q6 Matrices Matrix Entry and Coefficient Identities View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Show that $s_{qq}' + s_{pp}' = s_{qq} + s_{pp}$.
Q7 Matrices Matrix Entry and Coefficient Identities View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Express the coefficients $s_{ij}'$ of $S'$ in terms of those of $S$.
Q8 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
We seek an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ for which we have $s_{pq}' = 0$.
(a) Show that $s_{pq}' = 0$ if and only if $t = \tan\theta$ satisfies the equation $$t^2 + \frac{s_{pp} - s_{qq}}{s_{pq}} t - 1 = 0 \tag{1}$$
(b) Show that this equation admits one solution $t_0 \in ]-1, 1]$ and another $t_1 \notin ]-1, 1]$. What is the relationship between the angles $\theta_0$ and $\theta_1$ that correspond to these roots?
(c) In all that follows, we choose one of the two roots $t$ of equation (1). We thus have $s_{pq}' = 0$. A more precise choice will be made starting from question 12. Verify that $s_{pp}' - s_{pp} = t s_{pq}$; establish an analogous formula for $s_{qq}' - s_{qq}$.
(d) We decompose $S$ in the form $S = D + E$ with $D$ diagonal and $E$ with zero diagonal. We similarly decompose $S' = D' + E'$. Calculate $\|E'\|^2$ in terms of $\|E\|^2$ and $\left(s_{pq}\right)^2$.
(e) By justifying that $\|S'\| = \|S\|$, deduce an expression for $\|D'\|^2$ in terms of $\|D\|^2$ and $\left(s_{pq}\right)^2$.
Q9 Matrices Matrix Entry and Coefficient Identities View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Show that the coefficients of $S'$ are expressed uniquely in terms of those of $S$ and the root ($t_0$ or $t_1$) that we have chosen.
Q10 Matrices Matrix Norm, Convergence, and Inequality View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Suppose in this question that $s_{pq}$ is the coefficient of largest absolute value in $E$.
(a) Show that $\|E'\| \leqslant \rho \|E\|$ where $\rho < 1$ is a constant that we will make explicit.
(b) If we choose the root $t_0$, show furthermore that $\|D' - D\| \leqslant \|E\|$.
Q11 Matrices Matrix Entry and Coefficient Identities View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
By calculating $\left(s_{qq}' - s_{pp}'\right)^2 - \left(s_{qq} - s_{pp}\right)^2$, show that $$\left|s_{qq}' - s_{pp}'\right| \geqslant \left|s_{qq} - s_{pp}\right|$$
Q12 Matrices Matrix Entry and Coefficient Identities View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
From now on, and until the end of the problem, we choose the root $t_0$ of (1) and thus the angle $\theta_0$, mentioned in Question 8b.
(a) Show that $s_{pp} - s_{pp}'$ and $s_{qq}' - s_{qq}$ have the same sign as $s_{qq} - s_{pp}$.
(b) If $1 \leqslant i \leqslant n$, show that $$\left|s_{ii} - s_{qq}'\right| + \left|s_{ii} - s_{pp}'\right| - \left|s_{ii} - s_{pp}\right| - \left|s_{ii} - s_{qq}\right| \geqslant 0$$
Q13 Matrices Matrix Norm, Convergence, and Inequality View
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$. From now on we choose the root $t_0$ of (1).
We define $$R = \sum_{i,j=1}^{n} \left|s_{jj} - s_{ii}\right| \quad \text{and} \quad R' = \sum_{i,j=1}^{n} \left|s_{jj}' - s_{ii}'\right|$$
Show that $$R' - R \geqslant 2\left(\left|s_{qq}' - s_{qq}\right| + \left|s_{pp}' - s_{pp}\right|\right) = 2\sum_{i=1}^{n} \left|s_{ii}' - s_{ii}\right|$$
Q14 Matrices Matrix Norm, Convergence, and Inequality View
In the Jacobi algorithm, we start with a matrix $\Sigma \in \mathbf{S}_n$ and construct a sequence of symmetric matrices $\Sigma^{(m)}$, whose coefficients are denoted $\sigma_{ij}^{(m)}$, as follows:
  • We set $\Sigma^{(0)} = \Sigma$.
  • When $\Sigma^{(m)}$ is known, we choose a pair $(p_m, q_m)$ with $p_m < q_m$.
  • We then apply the calculations from Part 2 to the matrix $S = \Sigma^{(m)}$ and the pair $(p,q) = (p_m, q_m)$: we form the matrix $S'$ studied in this part, and call it $\Sigma^{(m+1)}$.

We define $$R_m = \sum_{i,j=1}^{n} \left|\sigma_{jj}^{(m)} - \sigma_{ii}^{(m)}\right|, \quad \varepsilon_m = \sum_{i=1}^{n} \left|\sigma_{ii}^{(m+1)} - \sigma_{ii}^{(m)}\right|$$
Verify that $R_{m+1} - R_m \geqslant 2\varepsilon_m$. Deduce that the series $\sum_{m=1}^{\infty} \varepsilon_m$ is convergent.
Q15 Matrices Matrix Norm, Convergence, and Inequality View
In the Jacobi algorithm, we start with a matrix $\Sigma \in \mathbf{S}_n$ and construct a sequence of symmetric matrices $\Sigma^{(m)}$, whose coefficients are denoted $\sigma_{ij}^{(m)}$. We decompose $\Sigma^{(m)}$ in the form $D^{(m)} + E^{(m)}$ where $D^{(m)}$ is diagonal and $E^{(m)}$ has zero diagonal.
Show that the sequence $\left(D^{(m)}\right)_{m \in \mathbb{N}}$ is convergent. We denote its limit by $D$.
Q16 Matrices Diagonalizability and Similarity View
Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$.
Show that $A^{(m)}$ is similar to $A^{(0)}$.
Q17 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$. We further assume that this sequence converges to a diagonal matrix $D$. If $P_m$ denotes the characteristic polynomial of $A^{(m)}$, show that the coefficients of $P_m$ converge to those of the characteristic polynomial of $D$ when $m \rightarrow +\infty$.
Deduce that the characteristic polynomial of $D$ is equal to that of $A^{(0)}$.
Q18 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$, and that this sequence converges to a diagonal matrix $D$.
Finally, show that the diagonal entries of $D$ are the eigenvalues of $A^{(0)}$. What can be said about their multiplicities?
Q19 Matrices Matrix Norm, Convergence, and Inequality View
In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words, $$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$
Show that as $m \rightarrow +\infty$, the sequence $\left(\Sigma^{(m)}\right)_{m \in \mathbb{N}}$ converges to the diagonal matrix $D$.
Q20 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words, $$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$
Show then that the diagonal coefficients of $D$ are the eigenvalues of $\Sigma$.
Q21 Matrices Matrix Norm, Convergence, and Inequality View
In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words, $$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$
Let $m \in \mathbb{N}$. Show that $$\left\|D - D^{(m)}\right\| \leqslant \frac{\rho^m}{1 - \rho} \left\|E^{(0)}\right\|$$
Q22 Bivariate data View
Based on your answers to the previous questions, give your opinion on the speed of convergence of the $d_{ii}^{(m)}$ to the eigenvalues of $\Sigma$.