grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2025 polytechnique-maths-a__mp

35 maths questions

Q1 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $V$ be a finite-dimensional vector space, let $h$ be an endomorphism of $V$ and let $W$ be a subspace stable by $h$. We denote by $h_W$ the endomorphism of $W$ induced by $h$, that is $h_W : W \rightarrow W$, $v \mapsto h(v)$. Prove that if $h$ is diagonalizable, then $h_W$ is also diagonalizable.
Q2a Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View
For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$ Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.
Q2b Invariant lines and eigenvalues and vectors Annihilating or minimal polynomial and spectral deductions View
Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.
Q2c Invariant lines and eigenvalues and vectors Annihilating or minimal polynomial and spectral deductions View
Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ One may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.
Q6c Groups Subgroup and Normal Subgroup Properties View
Let $W'$ be the subspace of $V$ defined by $$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$ Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.
Q6d Groups Decomposition and Basis Construction View
We denote $G_v = \varphi(u^r(v))$. Prove that there exists an element $F_v$ of $\mathcal{D}$ such that $$G_v = \xi^r(F_v).$$
Q6e Groups Decomposition and Basis Construction View
Let $P$ be a polynomial and let $w$ be an element of $W$. Prove that if $P(u)(v) = w$, then $P(\xi)(F_v) = \varphi(w)$.
Q6f Groups Decomposition and Basis Construction View
Let $x$ be an element of $W'$. Let $P$ be a polynomial and let $w$ be an element of $W$ such that $x = P(u)(v) + w$. Prove that the element $\varphi'(x) = P(\xi)(F_v) + \varphi(w)$ depends only on $x$ and not on the choice of $P$ and $w$. Verify then that the application $\varphi'$ thus defined is an extension of $\varphi$ to $W'$ compatible with $u$ (it is not asked to verify that $\varphi'$ is linear, which we will admit).
Q7 Groups Existence Proof View
Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.
Q8a Groups Linear Transformation and Endomorphism Properties View
We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero. Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.
Q8b Groups Existence Proof View
Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
Q8c Groups Proof of Set Membership, Containment, or Structural Property View
Verify that the image of $\psi$ is contained in the kernel of $\xi^n$.
Q8d Groups Proof of Stability or Invariance View
Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.
Q9 Groups Linear Transformation and Endomorphism Properties View
Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.
Q10 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. One may use question $2^\circ$.
Q11a Matrices Diagonalizability and Similarity View
Prove that $h$ is diagonalizable.
Q11b Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \mathrm{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.
Q11c Invariant lines and eigenvalues and vectors Matrix Algebra and Product Properties View
Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.
Q12a Linear transformations Projection and Orthogonality View
Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$. Verify that $u$ and $p$ commute.
Q12b Linear transformations Projection and Orthogonality View
We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
Q12c Linear transformations Projection and Orthogonality View
Deduce that $\bar{p}$ is a projector and that its image is $W$.
Q12d Linear transformations Projection and Orthogonality View
Prove carefully that $\bar{p}$ commutes with $u$ and $h$.
Q12e Linear transformations Projection and Orthogonality View
Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.
Q13a Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.
Q13b Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable.
Q14a Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
In this question, we further assume that $N = 4$ and $\ker(h - \mathrm{id}_V) = \{0\}$. Verify that $u^3 = 0$.
Q14b Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the graded version of the decomposition theorem.
Q14c Proof Determinant and Rank Computation View
Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.
Q15 Proof Diagonalizability and Similarity View
Let $(A, B)$ and $(A', B')$ be two pairs of matrices in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$. Prove that the following conditions are equivalent: (i) $(A, B)$ and $(A', B')$ are simultaneously equivalent; (ii) there exist $P \in \mathrm{GL}_m(\mathbb{C})$ and $Q \in \mathrm{GL}_n(\mathbb{C})$ such that $A' = QAP^{-1}$ and $B' = PBQ^{-1}$; (iii) there exists $R \in \mathrm{GL}_{m+n}(\mathbb{C})$ such that $M_{A',B'} = RM_{A,B}R^{-1}$ and $H = RHR^{-1}$.
Q16a Matrices Matrix Algebra and Product Properties View
Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.
Q16b Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.
Q16c Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.
Q17 Proof Diagonalizability and Similarity View
In this question, we assume that $M$ is nilpotent. Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form $$\left(\begin{array}{cc} 0_r & B_0 \\ A_0 & 0_s \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{array}\right),$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{s \times (s+1)} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \left(\begin{array}{cccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(r+1) \times r} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{r \times (r+1)}.$$
Q18a Proof Determinant and Rank Computation View
In this question, we assume that $M$ is invertible. Prove that $m = n$ and that $A$ and $B$ are invertible.
Q18b Proof Diagonalizability and Similarity View
Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\left(\begin{array}{cc} 0_r & B_1 \\ A_1 & 0_r \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{array}\right),$$ where $$A_1 = \mathrm{I}_r \quad \text{and} \quad B_1 = \lambda \mathrm{I}_r + J_r$$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.