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2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
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2017
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2016
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2015
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2014
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2013
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2012
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2011
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2010
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2019 centrale-maths1__official

37 maths questions

Q1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.
Q2 Matrices Diagonalizability and Similarity View
Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M^{\top}$ is diagonalisable if and only if $M$ is diagonalisable.
Q3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $\left(a_0, a_1, \ldots, a_{n-1}\right) \in \mathbb{K}^n$ and $Q(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_0$. We consider the matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Determine as a function of $Q$ the characteristic polynomial of $C_Q$.
Q4 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
Let $\left(a_0, a_1, \ldots, a_{n-1}\right) \in \mathbb{K}^n$ and $Q(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_0$. We consider the companion matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Let $\lambda$ be an eigenvalue of $C_Q^{\top}$. Determine the dimension and a basis of the associated eigenspace.
Q5 Matrices Linear Transformation and Endomorphism Properties View
Show that $f$ is cyclic if and only if there exists a basis $\mathcal{B}$ of $E$ in which the matrix of $f$ is of the form $C_Q$, where $Q$ is a monic polynomial of degree $n$.
Q6 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $f$ be a cyclic endomorphism. Show that $f$ is diagonalisable if and only if $\chi_f$ is split over $\mathbb{K}$ and has all its roots simple.
Q7 Groups Automorphism and Endomorphism Structure View
Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.
Q8 Groups Decomposition and Basis Construction View
Let $x$ be a nonzero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$
Q9 Groups Subgroup and Normal Subgroup Properties View
Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.
Q10 Roots of polynomials Divisibility and minimal polynomial arguments View
Show that: $X^p + \alpha_{p-1}X^{p-1} + \cdots + \alpha_0$ divides the polynomial $\chi_f$.
Q11 Roots of polynomials Divisibility and minimal polynomial arguments View
Prove that $\chi_f(f)$ is the zero endomorphism.
Q12 Matrices Linear Transformation and Endomorphism Properties View
We assume that $f$ is a nilpotent endomorphism of $E$. We denote by $r$ the smallest natural integer such that $f^r = 0$. Show that $f$ is cyclic if and only if $r = n$. Specify the companion matrix.
Q13 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$.
Show that the vector subspaces $F_k$ are stable under $f$ and that $E = F_1 \oplus \cdots \oplus F_p$.
Q14 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$, and we denote by $\varphi_k$ the endomorphism induced by $f - \lambda_k \operatorname{Id}$ on the vector subspace $F_k$,
$$\varphi_k : \left\lvert\, \begin{aligned} & F_k \rightarrow F_k, \\ & x \mapsto f(x) - \lambda_k x. \end{aligned} \right.$$
Justify that $\varphi_k$ is a nilpotent endomorphism of $F_k$.
Q15 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. For $k \in \llbracket 1, p \rrbracket$, $\varphi_k$ is a nilpotent endomorphism of $F_k$, and $\nu_k$ denotes the smallest natural integer such that $\varphi_k^{\nu_k} = 0$. Why do we have $\nu_k \leqslant \operatorname{dim}(F_k)$?
Q16 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, $\varphi_k$ is a nilpotent endomorphism of $F_k$, and $\nu_k$ denotes the smallest natural integer such that $\varphi_k^{\nu_k} = 0$.
Show, with the proposed hypothesis, that for all $k \in \llbracket 1, p \rrbracket$, we have $\nu_k = m_k$.
Q17 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, $\varphi_k$ is a nilpotent endomorphism of $F_k$ with $\nu_k = m_k$.
Specify the dimension of $F_k$ for $k \in \llbracket 1, p \rrbracket$, then deduce the existence of a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix, these blocks belonging to $\mathcal{M}_{m_k}(\mathbb{C})$ and being of the form
$$\left(\begin{array}{cccccc} \lambda_k & 0 & \cdots & \cdots & \cdots & 0 \\ 1 & \lambda_k & \ddots & & & \vdots \\ 0 & 1 & \lambda_k & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & \lambda_k & 0 \\ 0 & \cdots & \cdots & 0 & 1 & \lambda_k \end{array}\right)$$
Q18 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and that there exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with blocks of size $m_k$ of the form
$$\left(\begin{array}{cccccc} \lambda_k & 0 & \cdots & \cdots & \cdots & 0 \\ 1 & \lambda_k & \ddots & & & \vdots \\ 0 & 1 & \lambda_k & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & \lambda_k & 0 \\ 0 & \cdots & \cdots & 0 & 1 & \lambda_k \end{array}\right)$$
We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Determine the polynomials $Q \in \mathbb{C}[X]$ such that $Q(f)(x_0) = 0$.
Q19 Matrices Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and that $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$ where $(u_1, \ldots, u_n)$ is the basis described in Q17. Justify that $f$ is cyclic.
Q20 Matrices Matrix Algebra and Product Properties View
We call the commutant of $f$ the set $\mathcal{C}(f) = \{g \in \mathcal{L}(E) \mid f \circ g = g \circ f\}$. Show that $\mathcal{C}(f)$ is a subalgebra of $\mathcal{L}(E)$.
Q21 Matrices Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$. Justify the existence of $\lambda_0, \lambda_1, \ldots, \lambda_{n-1}$ of $\mathbb{K}$ such that
$$g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$$
Q22 Matrices Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$ with $g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$. Show then that $g \in \mathbb{K}[f]$.
Q23 Matrices Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic. Establish that $g \in \mathcal{C}(f)$ if and only if there exists a polynomial $R \in \mathbb{K}_{n-1}[X]$ such that $g = R(f)$.
Q24 Proof Proof of Set Membership, Containment, or Structural Property View
Show that if the union of a finite number of vector subspaces $F_1, \ldots, F_r$ of $E$ is a vector subspace, then one of the vector subspaces $F_i$ contains all the others.
Q25 Matrices Linear Transformation and Endomorphism Properties View
We denote by $d$ the degree of $\pi_f$. Justify the existence of a vector $x_1$ of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free.
Q26 Matrices Linear Transformation and Endomorphism Properties View
We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$. Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.
Q27 Matrices Linear Transformation and Endomorphism Properties View
We set $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$, and we denote by $\psi_1$ the endomorphism induced by $f$ on the vector subspace $E_1$,
$$\psi_1 : \left\lvert\, \begin{aligned} & E_1 \rightarrow E_1, \\ & x \mapsto f(x). \end{aligned} \right.$$
Justify that $\psi_1$ is cyclic.
Q28 Matrices Linear Transformation and Endomorphism Properties View
We complete, if necessary, $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. We denote by $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$.
Show that $F$ is stable under $f$ and that $E_1$ and $F$ are in direct sum.
Q29 Matrices Linear Transformation and Endomorphism Properties View
We complete, if necessary, $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. Let $\Psi$ be the linear map from $E$ to $\mathbb{K}^d$ defined, for all $x \in E$, by
$$\Psi(x) = \left(\Phi\left(f^i(x)\right)\right)_{0 \leqslant i \leqslant d-1} = \left(\Phi(x), \Phi(f(x)) \ldots, \Phi\left(f^{d-1}(x)\right)\right)$$
Show that $\Psi$ induces an isomorphism between $E_1$ and $\mathbb{K}^d$.
Q30 Matrices Linear Transformation and Endomorphism Properties View
Using the notation of Q28 and Q29, show that $E = E_1 \oplus F$.
Q31 Matrices Linear Transformation and Endomorphism Properties View
Deduce that there exist $r$ vector subspaces of $E$, denoted $E_1, \ldots, E_r$, all stable under $f$ such that:
  • $E = E_1 \oplus \cdots \oplus E_r$;
  • for all $1 \leqslant i \leqslant r$, the endomorphism $\psi_i$ induced by $f$ on the vector subspace $E_i$ is cyclic;
  • if we denote by $P_i$ the minimal polynomial of $\psi_i$, then $P_{i+1}$ divides $P_i$ for all integer $i$ such that $1 \leqslant i \leqslant r-1$.
Q32 Matrices Linear Transformation and Endomorphism Properties View
Show that the dimension of $\mathcal{C}(f)$ is greater than or equal to $n$.
Q33 Matrices Linear Transformation and Endomorphism Properties View
We assume that $f$ is an endomorphism such that the algebra $\mathcal{C}(f)$ is equal to $\mathbb{K}[f]$. Show that $f$ is cyclic.
Q34 Matrices Projection and Orthogonality View
In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Let $f' \in \mathrm{O}(E)$ having the same characteristic polynomial as $f$. Show that there exist orthonormal bases $\mathcal{B}$ and $\mathcal{B}'$ of $E$ for which the matrix of $f$ in $\mathcal{B}$ is equal to the matrix of $f'$ in $\mathcal{B}'$.
Q35 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Deduce that $f$ is orthocyclic if and only if $\chi_f = X^n - 1$ or $\chi_f = X^n + 1$.
Q36 Matrices Projection and Orthogonality View
In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.
Q37 Matrices Linear Transformation and Endomorphism Properties View
In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix). Let $f$ be a nilpotent endomorphism of $E$. Deduce that $f$ is orthocyclic if and only if
$$f \text{ has rank } n-1 \quad \text{and} \quad \forall x, y \in (\ker f)^{\perp}, \quad (f(x) \mid f(y)) = (x \mid y).$$