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
2012 centrale-maths2__mp

27 maths questions

QI.A Sequences and Series Recurrence Relations and Sequence Properties View
Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.
We recall that this implies two things:
  • on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
  • on the other hand, the elements of $J_x$ are the multiples of $B$.
By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.
QI.B.1 Sequences and Series Recurrence Relations and Sequence Properties View
In $\mathbb{K}^{\mathbb{N}}$, what are the linear recurrent sequences of order 0? of order 1?
What are the sequences in $\mathbb{K}^{\mathbb{N}}$ whose minimal polynomial is $(X-1)^2$?
QI.B.2 Sequences and Series Recurrence Relations and Sequence Properties View
We consider the sequence $x$ defined by $x_0 = 0, x_1 = -1, x_2 = 2$ and by the linear recurrence relation of order 3: $\forall n \in \mathbb{N}, x_{n+3} = -3x_{n+2} - 3x_{n+1} - x_n$.
Determine the minimal polynomial (and thus the minimal order) of the sequence $x$.
QI.C.1 Sequences and Series Recurrence Relations and Sequence Properties View
Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Prove that $\mathcal{R}_A(\mathbb{K})$ is a vector subspace of dimension $p$ of $\mathbb{K}^{\mathbb{N}}$ and that it is stable under $\sigma$ (we do not ask here to determine a basis of $\mathcal{R}_A(\mathbb{K})$, as this is the object of the following questions).
QI.C.2 Sequences and Series Recurrence Relations and Sequence Properties View
Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Determine $\mathcal{R}_A(\mathbb{K})$ when $A = X^p$ (with $p \geqslant 1$) and give a basis for it.
QI.C.3 Sequences and Series Recurrence Relations and Sequence Properties View
Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$.
We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$.
a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension.
b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.
QI.D Sequences and Series Recurrence Relations and Sequence Properties View
In this question, we assume that the polynomial $A$ is split over $\mathbb{K}$. More precisely, we write $A = X^{m_0} \prod_{k=1}^{d} (X - \lambda_k)^{m_k}$, where:
  • the scalars $\lambda_1, \lambda_2, \ldots, \lambda_d$ are the distinct non-zero roots of $A$ in $\mathbb{K}$, and $m_1, m_2, \ldots, m_d$ are their respective multiplicities (greater than or equal to 1). If $A$ has no non-zero root, we adopt the convention that $d = 0$ and that $\prod_{k=1}^{d}(X-\lambda_k)^{m_k} = 1$;
  • the integer $m_0$ is the multiplicity of 0 as a possible root of $A$. If 0 is not a root of $A$, we adopt the convention $m_0 = 0$.
With these notations, we have $\sum_{k=0}^{d} m_k = \deg A = p$.
Using the kernel decomposition theorem, show that $\mathcal{R}_A(\mathbb{K})$ is the set of sequences $x = (x_n)_{n \geqslant 0}$ in $\mathbb{K}^{\mathbb{N}}$ such that: $$\forall n \geqslant m_0, \quad x_n = \sum_{k=1}^{d} Q_k(n) \lambda_k^n$$ where, for all $k$ in $\{1, \ldots, d\}$, $Q_k$ is in $\mathbb{K}[X]$ with $\deg Q_k < m_k$.
Remark: if $d = 0$, the sum $\sum_{k=1}^{d} Q_k(n)\lambda_k^n$ is by convention equal to 0.
QII.A.1 Matrices Determinant and Rank Computation View
Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$.
Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.
QII.A.2 Matrices Determinant and Rank Computation View
Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that if $n \geqslant p$, the map $\varphi_n : \left\{ \begin{array}{l} \mathcal{R}_B(\mathbb{K}) \rightarrow \mathbb{K}^n \\ v \mapsto (v_0, \ldots, v_{n-1}) \end{array} \right.$ is injective.
Deduce from this that if $n \geqslant p$, then $\operatorname{rang}(H_n(x)) = p$.
Remark: it is clear that this result remains true if $p = 0$ (since the sequence $x$ and the matrices $H_n(x)$ are zero).
QII.B.1 Matrices Linear Transformation and Endomorphism Properties View
Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
QII.B.2 Sequences and Series Recurrence Relations and Sequence Properties View
Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. The kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
With these notations, show that the minimal polynomial of $x$ is $B = X^p + b_{p-1}X^{p-1} + \cdots + b_1 X + b_0$.
QII.C.1 Sequences and Series Algorithmic/Computational Implementation for Sequences and Series View
We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
In the computer language of your choice (which you will specify), write a procedure (or function) with parameter a natural number $n$ and returning the list (or sequence, or vector) of $x_k$ for $0 \leqslant k \leqslant n$.
QII.C.2 Sequences and Series Recurrence Relations and Sequence Properties View
We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Specify the rank of $H_n(x)$ for any integer $n$ in $\mathbb{N}^*$ and indicate the minimal order of the sequence $x$.
QII.C.3 Sequences and Series Recurrence Relations and Sequence Properties View
We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Determine the minimal recurrence relation of the sequence $x$.
QII.C.4 Sequences and Series Recurrence Relations and Sequence Properties View
We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Give a formula allowing for any $n \geqslant 1$ to directly compute $x_n$.
QII.C.5 Sequences and series, recurrence and convergence Closed-form expression derivation View
We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
We decide to modify only the value of $x_0$, by setting this time $x_0 = \frac{1}{2}$.
With this modification, quickly redo the study of questions II.C.2 and II.C.3.
QIII.A.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3. We say that a matrix $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ in $\mathcal{M}_n(\mathbb{R})$ is a Hankel matrix if there exists $a = (a_0, \ldots, a_{2n-2}) \in \mathbb{R}^{2n-1}$ such that for all $i$ and $j$ in $\{1, \ldots, n\}$, $m_{i,j} = a_{i+j-2}$. Such a matrix is denoted $M = H(a)$.
Show that if $M$ is a Hankel matrix of size $n$ then it admits $n$ real eigenvalues $\lambda_1, \ldots, \lambda_n$ (each repeated as many times as its multiplicity) which can be ordered in decreasing order $\lambda_1 \geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_n$.
QIII.A.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3. We say that a matrix $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ in $\mathcal{M}_n(\mathbb{R})$ is a Hankel matrix if there exists $a = (a_0, \ldots, a_{2n-2}) \in \mathbb{R}^{2n-1}$ such that for all $i$ and $j$ in $\{1, \ldots, n\}$, $m_{i,j} = a_{i+j-2}$. Such a matrix is denoted $M = H(a)$.
Show that if $\lambda \in \mathbb{R}^*$ then the $n$-tuple $(\lambda, \ldots, \lambda)$ is not the ordered $n$-tuple of eigenvalues of a Hankel matrix of size $n$.
QIII.B.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $$\sum_{i=1}^{n} \lambda_i = \sum_{k=0}^{n-1} a_{2k} \quad \text{and} \quad \sum_{i=1}^{n} \lambda_i^2 = \sum_{k=0}^{n-1} (k+1) a_k^2 + \sum_{k=n}^{2n-2} (2n-k-1) a_k^2$$
QIII.B.2 Matrices Matrix Norm, Convergence, and Inequality View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $\langle v, w \rangle = \sum_{i=1}^{n} \lambda_i$ and $\|v\|^2 \leqslant \sum_{i=1}^{n} \lambda_i^2$.
QIII.B.3 Matrices Matrix Norm, Convergence, and Inequality View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
We define two vectors $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$ of $\mathbb{R}^n$ by $$\begin{cases} v_i = \sqrt{2i-1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2i-1}} & \text{if } i \in \{1, \ldots, p\} \\ v_i = \sqrt{2n-2i+1}\, a_{2(i-1)} \text{ and } w_i = \dfrac{1}{\sqrt{2n-2i+1}} & \text{if } i \in \{p+1, \ldots, n\} \end{cases}$$ Finally, we set $K_n = n - \|w\|^2$.
Show that $\displaystyle\sum_{1 \leqslant i < j \leqslant n} (\lambda_i - \lambda_j)^2 = n \sum_{i=1}^{n} \lambda_i^2 - \langle v, w \rangle^2$ and deduce the inequality: $$\sum_{1 \leqslant i < j \leqslant n} (\lambda_i - \lambda_j)^2 \geqslant K_n \sum_{i=1}^{n} \lambda_i^2 \tag{III.1}$$
QIII.B.4 Matrices Matrix Norm, Convergence, and Inequality View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Verify that if $n = 3$, condition III.1 is equivalent to: $2(\lambda_1^2 + \lambda_2^2 + \lambda_3^2) \geqslant 3(\lambda_1\lambda_2 + \lambda_1\lambda_3 + \lambda_2\lambda_3)$.
QIII.C.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$ all other coefficients of $B$ being zero.
Determine the ordered spectrum of matrix $B$.
QIII.C.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
We admit the following result: if $A$ and $B$ are two matrices of $\mathcal{S}_n(\mathbb{R})$ whose respective eigenvalues (with possible repetitions) are $\alpha_1 \geqslant \ldots \geqslant \alpha_n$ and $\beta_1 \geqslant \ldots \geqslant \beta_n$ then $$\sum_{i=1}^{n} \alpha_i \beta_{n+1-i} \leqslant \operatorname{tr}(AB) \leqslant \sum_{i=1}^{n} \alpha_i \beta_i$$
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$ all other coefficients of $B$ being zero.
Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
Establish that $$\lambda_1 - \lambda_{n-1} - 2\lambda_n \geqslant 0 \quad \text{and} \quad 2\lambda_1 + \lambda_2 - \lambda_n \geqslant 0 \tag{III.3}$$
QIII.D.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
Calculate the eigenvalues of $M$ (without trying to order them).
QIII.D.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
Explicitly express $a, b, c$ (with $b \geqslant 0$) as functions of $\lambda_1, \lambda_2, \lambda_3$, such that $\operatorname{Spo}(M) = (\lambda_1, \lambda_2, \lambda_3)$.
QIII.D.3 Matrices True/False or Multiple-Select Conceptual Reasoning View
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0 \tag{III.3}$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
What can be deduced from the previous result, regarding condition III.3 in the case $n = 3$? Using an ordered triplet $(\lambda, 1, 1)$, show that for $n = 3$, condition III.1 is not sufficient.