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
2018 centrale-maths1__psi

44 maths questions

Q1 Matrices Structured Matrix Characterization View
Show that $\operatorname{Toep}_{n}(\mathbb{C})$ is a vector subspace of $\mathcal{M}_{n}(\mathbb{C})$. Give a basis for it and specify its dimension.
Q2 Matrices Matrix Algebra and Product Properties View
Show that if two matrices $A$ and $B$ commute $(AB = BA)$ and if $P$ and $Q$ are two polynomials of $\mathbb{C}[X]$, then $P(A)$ and $Q(B)$ commute.
Q3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Give the characteristic polynomial of $A$.
Q4 Matrices Diagonalizability and Similarity View
Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Discuss, depending on the values of $(a, b, c)$, the diagonalizability of $A$.
Q5 Matrices Diagonalizability and Similarity View
Let $M = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix of $\mathcal{M}_{2}(\mathbb{C})$. Show that $M$ is similar to a matrix of type $\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array}\right)$ or of type $\left(\begin{array}{cc} \alpha & \gamma \\ 0 & \alpha \end{array}\right)$, where $\alpha, \beta$ and $\gamma$ are complex numbers with $\alpha \neq \beta$.
Q6 Matrices Diagonalizability and Similarity View
Deduce that every matrix of $\mathcal{M}_{2}(\mathbb{C})$ is similar to a Toeplitz matrix.
Q7 Second order differential equations Second-order linear recurrence relation View
A tridiagonal matrix is a Toeplitz matrix of the form $T(0,\ldots,0,t_{-1},t_0,t_1,0,\ldots,0)$, i.e. a matrix of the form $$A_n(a,b,c) = \left(\begin{array}{cccc} a & b & & (0) \\ c & a & \ddots & \\ & \ddots & \ddots & b \\ (0) & & c & a \end{array}\right)$$ where $(a,b,c)$ are complex numbers. We fix $(a,b,c)$ three complex numbers such that $bc \neq 0$. Let $\lambda \in \mathbb{C}$ be an eigenvalue of $A_n(a,b,c)$ and $X = \left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array}\right) \in \mathbb{C}^n$ be an associated eigenvector.
Show that if we set $x_0 = 0$ and $x_{n+1} = 0$, then $(x_1, \ldots, x_n)$ are the terms of rank varying from 1 to $n$ of a sequence $(x_k)_{k \in \mathbb{N}}$ satisfying $x_0 = 0, x_{n+1} = 0$ and $$\forall k \in \mathbb{N}, \quad bx_{k+2} + (a-\lambda)x_{k+1} + cx_k = 0$$
Q8 Second order differential equations Second-order linear recurrence relation View
Recall the expression of the general term of the sequence $(x_k)_{k \in \mathbb{N}}$ as a function of the solutions of the equation $$bx^2 + (a-\lambda)x + c = 0 \tag{I.1}$$
Q9 Second order differential equations Second-order linear recurrence relation View
Using the conditions imposed on $x_0$ and $x_{n+1}$, show that (I.1) admits two distinct solutions $r_1$ and $r_2$.
Q10 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Show that $r_1$ and $r_2$ are nonzero and that $r_1/r_2$ belongs to $\mathbb{U}_{n+1}$.
Q11 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that $$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$
Q12 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.
Q13 Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View
Conclude that $A_n(a,b,c)$ is diagonalizable and give its eigenvalues.
Q14 Matrices Matrix Power Computation and Application View
We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Calculate $M_n^2, \ldots, M_n^n$. Show that $M_n$ is invertible and give an annihilating polynomial of $M_n$.
Q15 Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View
We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Justify that $M_n$ is diagonalizable. Specify its eigenvalues (expressed using $\omega_n$) and give a basis of eigenvectors of $M_n$.
Q16 Matrices Diagonalizability and Similarity View
We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
We set $\Phi_n = \left(\omega_n^{(p-1)(q-1)}\right)_{1 \leqslant p,q \leqslant n} \in \mathcal{M}_n(\mathbb{C})$. Justify that $\Phi_n$ is invertible and give without calculation the value of the matrix $\Phi_n^{-1} M_n \Phi_n$.
Q17 Matrices Structured Matrix Characterization View
Let $A$ be a circulant matrix. Give a polynomial $P \in \mathbb{C}[X]$ such that $A = P(M_n)$.
Q18 Matrices Structured Matrix Characterization View
Conversely, if $P \in \mathbb{C}[X]$, show, using a Euclidean division of $P$ by a suitably chosen polynomial, that $P(M_n)$ is a circulant matrix.
Q19 Matrices Structured Matrix Characterization View
Show that the set of circulant matrices is a vector subspace of $\operatorname{Toep}_n(\mathbb{C})$, stable under multiplication and transposition.
Q20 Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View
Show that every circulant matrix is diagonalizable. Specify its eigenvalues and a basis of eigenvectors.
Q21 Matrices Linear Transformation and Endomorphism Properties View
Show that if $M$ is in $\mathcal{M}_n(\mathbb{C})$, then the following propositions are equivalent:
i. there exists $x_0$ in $\mathbb{C}^n$ such that $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ is a basis of $\mathbb{C}^n$;
ii. $M$ is similar to the matrix $C(a_0, \ldots, a_{n-1})$ defined by $$C(a_0, \ldots, a_{n-1}) = \left(\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_0 \\ 1 & \ddots & & \vdots & a_1 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & a_{n-1} \end{array}\right)$$ where $(a_0, \ldots, a_{n-1})$ are complex numbers.
Q22 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $M$ be in $\mathcal{M}_n(\mathbb{C})$. We assume that $f_M$ is diagonalizable. We denote by $(\lambda_1, \ldots, \lambda_n)$ its eigenvalues (not necessarily distinct) and by $(e_1, \ldots, e_n)$ a basis of eigenvectors associated with these eigenvalues. Let $u = \sum_{i=1}^{n} u_i e_i$ be a vector of $\mathbb{C}^n$ where $(u_1, \ldots, u_n)$ are $n$ complex numbers.
Give a necessary and sufficient condition on $(u_1, \ldots, u_n, \lambda_1, \ldots, \lambda_n)$ for $(u, f_M(u), \ldots, f_M^{n-1}(u))$ to be a basis of $\mathbb{C}^n$.
Q23 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Deduce a necessary and sufficient condition for a diagonalizable endomorphism to be cyclic. Then characterize its cyclic vectors.
Q24 Roots of polynomials Eigenvalue-root connection for matrices or linear operators View
Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. Let $\lambda$ be a complex number. By discussing in $\mathbb{C}^n$ the system $C(a_0, \ldots, a_{n-1})X = \lambda X$, show that $\lambda$ is an eigenvalue of $C(a_0, \ldots, a_{n-1})$ if and only if $\lambda$ is a root of a polynomial of $\mathbb{C}[X]$ to be specified.
Q25 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. If $\lambda$ is a root of the polynomial identified in Q24, determine the eigenspace of $C(a_0, \ldots, a_{n-1})$ associated with the eigenvalue $\lambda$ and specify its dimension.
Q26 Matrices Diagonalizability and Similarity View
Deduce a necessary and sufficient condition for a cyclic matrix to be diagonalizable.
Q27 Matrices Linear Transformation and Endomorphism Properties View
Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $P \in \mathbb{C}[X]$. Show that $P(f_M) \in \mathcal{C}(f_M)$, where $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$.
Q28 Matrices Linear Transformation and Endomorphism Properties View
Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $g \in \mathcal{C}(f_M)$. Show that there exist $(\alpha_0, \ldots, \alpha_{n-1}) \in \mathbb{C}^n$ such that $g = \alpha_0 Id_{\mathbb{C}^n} + \alpha_1 f_M + \cdots + \alpha_{n-1} f_M^{n-1}$. One may use the basis $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ and express $g(x_0)$ in this basis.
Q29 Matrices Linear Transformation and Endomorphism Properties View
Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. The set $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$ is sought to be shown to be the set of polynomials in $f_M$. Conclude.
Q30 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Give the eigenvalues of $N$ and the associated eigenspaces. Is it diagonalizable?
Q31 Matrices Linear Transformation and Endomorphism Properties View
Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Is the matrix $N$ cyclic?
Q32 Matrices Structured Matrix Characterization View
Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Show that the set of matrices that commute with $N$ is the set of lower triangular Toeplitz matrices.
Q33 Matrices Matrix Algebra and Product Properties View
Show that if $i$ and $j$ are in $\llbracket -n+1, n-1 \rrbracket$, if $A \in \Delta_i$ and $B \in \Delta_j$, then $AB \in \Delta_{i+j}$.
Q34 Matrices Matrix Algebra and Product Properties View
Deduce that if $A \in H_i$ and $B \in H_j$, then $AB \in H_{i+j}$.
Q35 Matrices Linear System and Inverse Existence View
Let $C$ be a nilpotent matrix. Show that $I_n + C$ is invertible and that $$\left(I_n + C\right)^{-1} = I_n - C + C^2 + \cdots + (-1)^{n-1} C^{n-1}$$
Q36 Proof Proof That a Map Has a Specific Property View
We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. Show that $P$ is invertible and that $P^{-1} \in \bigoplus_{p=0}^{n-1} \Delta_{p(k+1)}$.
Q37 Proof Direct Proof of a Stated Identity or Equality View
We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$.
Let $i \in \llbracket 0, k \rrbracket$ and $M \in \Delta_i$. Show that there exists $M'$ in $H_{k+1}$ such that $\varphi(M) = M + M'$.
Q38 Proof Direct Proof of a Stated Identity or Equality View
We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. The matrix $N$ being the matrix defined in III.A.4, show that there exists $N'$ in $H_{k+1}$ such that $$\varphi(N) = N + NC - CN + N'$$
Q39 Proof Direct Proof of a Stated Identity or Equality View
We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. Let $T$ be an upper triangular matrix. We set $A = N + T$, $B = \varphi(A)$. Show that $B \in H_{-1}$ and that $$\begin{cases} \forall i \in \llbracket -1, k-1 \rrbracket, \quad B^{(i)} = A^{(i)} \\ B^{(k)} = A^{(k)} + NC - CN \end{cases}$$
Q40 Matrices Linear Transformation and Endomorphism Properties View
We define the operators $$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$ Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.
Q41 Matrices Linear Transformation and Endomorphism Properties View
Show that $\mathcal{S}(\Delta_{k+1}) \subset \Delta_k$ and $\mathcal{S}^*(\Delta_k) \subset \Delta_{k+1}$.
Q42 Matrices Projection and Orthogonality View
We equip $\mathcal{M}_n(\mathbb{R})$ with its usual inner product defined by: $\forall (M_1, M_2) \in \mathcal{M}_n(\mathbb{R}), \langle M_1, M_2 \rangle = \operatorname{tr}({}^t M_1 M_2)$. We denote by $\mathcal{S}_{k+1}$ the restriction of $\mathcal{S}$ to $\Delta_{k+1}$ and $\mathcal{S}_k^*$ the restriction of $\mathcal{S}^*$ to $\Delta_k$.
Verify that for all $X$ in $\Delta_{k+1}$ and $Y$ in $\Delta_k$, $\langle \mathcal{S}_{k+1} X, Y \rangle = \langle X, \mathcal{S}_k^* Y \rangle$. Deduce that $\ker(\mathcal{S}_k^*)$ and $\operatorname{Im}(\mathcal{S}_{k+1})$ are orthogonal complements in $\Delta_k$, that is $$\Delta_k = \ker(\mathcal{S}_k^*) \oplus^{\perp} \operatorname{Im}(\mathcal{S}_{k+1})$$
Q43 Matrices Diagonalizability and Similarity View
Let $T$ be an upper triangular matrix, $A = N + T$ and $k \geqslant 0$. Show that $A$ is similar to a matrix $L$ whose diagonal coefficients of order $k$ are all equal and satisfying $\forall i \in \llbracket -1, k-1 \rrbracket, L^{(i)} = A^{(i)}$.
Q44 Matrices Diagonalizability and Similarity View
Deduce that every cyclic matrix is similar to a Toeplitz matrix.