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
2016 centrale-maths1__mp

41 maths questions

QI.A.1 Matrices Matrix Algebraic Properties and Abstract Reasoning View
For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Show that the map $A \mapsto N(A)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.
QI.A.2 Matrices Matrix Algebraic Properties and Abstract Reasoning View
For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Let $Q \in \mathrm{GL}_n(\mathbb{K})$. Show that $A \mapsto \|A\| = N\left(Q^{-1}AQ\right)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.
QI.B.1 Matrices Matrix Algebraic Properties and Abstract Reasoning View
We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. We want to show that $\lim_{m \rightarrow +\infty} A^m = 0$.
Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.
Show that $\widehat{T}$ is upper triangular and that we can choose $\delta$ so that $N(\widehat{T}) < 1$.
QI.B.2 Matrices Matrix Algebraic Properties and Abstract Reasoning View
We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.
With the choice of $\delta$ such that $N(\widehat{T}) < 1$, we set $Q = P\Delta$ and we equip $\mathcal{M}_n(\mathbb{C})$ with the norm $M \mapsto \|M\| = N\left(Q^{-1}MQ\right)$.
Show that $\|A\| < 1$ and deduce $\lim_{m \rightarrow +\infty} A^m = 0$.
QII.A Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that if there exists in $A$ a path from $i$ to $j$, with $i \neq j$, then there exists an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$. Similarly, show that if there exists in $A$ a circuit passing through $i$, then there exists an elementary circuit passing through $i$ of length $\ell \leqslant n$.
QII.B Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$. Let $m \geqslant 1$. Show the equivalence of the propositions:
  • there exists in $A$ a path with origin $i$, endpoint $j$, of length $m$;
  • the entry with indices $i, j$ of $A^m$ (denoted $a_{i,j}^{(m)}$) is strictly positive.
You may proceed by induction on the integer $m \geqslant 1$.
QII.C Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$, and let $\ell$ and $m$ be in $\mathbb{N}^*$. Show the equivalence of the propositions:
  • there exists in $A^m$ a path with origin $i$, endpoint $j$, of length $\ell$;
  • there exists in $A$ a path with origin $i$, endpoint $j$, of length $m\ell$.
QIII.A Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A$ be a primitive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that for all $i \neq j$ there exists in $A$ an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$, and that for all $i$ there exists in $A$ an elementary circuit passing through $i$ of length $\ell \leqslant n$.
QIII.B.1 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Give a simple example of a square matrix that is primitive but not strictly positive.
QIII.B.2 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $B > 0$ in $\mathcal{M}_n(\mathbb{R})$ and $x \geqslant 0$ in $\mathbb{R}^n$ with $x \neq 0$. Show that $Bx > 0$.
QIII.B.3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A$ be a primitive matrix and $m \in \mathbb{N}^*$ such that $A^m > 0$. Show that $\forall p \geqslant m, A^p > 0$. You may note, by justifying it, that none of the columns $c_1, c_2, \ldots, c_n$ of $A$ is zero.
QIII.B.4 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Prove that if $A$ is primitive, then $A^k$ is primitive for all $k \geqslant 1$.
QIII.B.5 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Show that the spectral radius of a primitive matrix is strictly positive.
QIII.C.1 Roots of polynomials Characteristic Polynomial of a Structured Matrix View
We define the matrix $W_n = (w_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$ by $w_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } i = n \text{ and } j \in \{1,2\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $W_n$ is $X^n - X - 1$.
Deduce that $W_n^{n^2-2n+1} = \sum_{k=1}^{n-1} \binom{n-2}{k-1} W_n^k$, then that $W_n^{n^2-2n+2} = I_n + W_n + \sum_{k=2}^{n-1} \binom{n-2}{k-2} W_n^k$.
QIII.C.2 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
We define the matrix $W_n = (w_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$ by $w_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } i = n \text{ and } j \in \{1,2\} \\ 0 & \text{in all other cases} \end{cases}$
Specify the shortest circuit passing through index 1 in the matrix $W_n$.
Deduce that the positive matrix $W_n^{n^2-2n+1}$ is not strictly positive.
QIII.C.3 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
We define the matrix $W_n = (w_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$ by $w_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } i = n \text{ and } j \in \{1,2\} \\ 0 & \text{in all other cases} \end{cases}$
Show that for all $i, j$ in $\llbracket 1, n \rrbracket$, with $i \neq j$, there exists in $W_n$ at least one path with origin $i$, endpoint $j$, and length less than or equal to $n-1$.
You may treat successively the two cases $1 \notin \{i,j\}$ and $1 \in \{i,j\}$.
Deduce that the matrix $W_n^{n^2-2n+2}$ is strictly positive and conclude.
QIII.D.1 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
Throughout this subsection, $A$ is a given primitive matrix in $\mathcal{M}_n(\mathbb{R})$. We denote by $\ell \in \llbracket 1, n \rrbracket$ the smallest length of an elementary circuit of $A$.
By contradiction, we suppose $\ell = n$.
Show that then all circuits of $A$ have length a multiple of $n$. Deduce that the matrices $A^{kn+1}$ (with $k \in \mathbb{N}$) have zero diagonal and reach a contradiction.
QIII.D.2 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
Throughout this subsection, $A$ is a given primitive matrix in $\mathcal{M}_n(\mathbb{R})$. According to what precedes, there exists in $A$ an elementary circuit $\mathcal{C}$ of length $\ell \leqslant n-1$. To simplify the exposition, and because it does not affect the generality of the problem, we assume that it is the circuit $1 \rightarrow 2 \rightarrow \ldots \rightarrow \ell-1 \rightarrow \ell \rightarrow 1$ (the remaining $n-\ell$ indices $\ell+1, \ell+2, \ldots, n$ being thus located ``outside'' the circuit $\mathcal{C}$).
We will show that $A^{n+\ell(n-2)}$ is strictly positive. For this, we are given $i$ and $j$ in $\llbracket 1, n \rrbracket$. Everything comes down to establishing that there exists in $A$ a path with origin $i$, endpoint $j$ and length $n + \ell(n-2)$.
a) Show that in $A$, we can form a path with origin $i$, of length $n-\ell$, whose endpoint is in $\{1, 2, \ldots, \ell\}$ (we will denote by $k$ this endpoint). You may treat the case $1 \leqslant i \leqslant \ell$, then the case $\ell+1 \leqslant i \leqslant n$.
b) State the reason why the first $\ell$ diagonal coefficients of $A^\ell$ (and in particular the $k$-th) are strictly positive. Show then that there exists a path of length $n-1$ in $A^\ell$ (that is, a path of length $\ell(n-1)$ in $A$) with origin $k$ and endpoint $j$.
c) Finally deduce $A^{n+\ell(n-2)} > 0$, then $A^{n^2-2n+2} > 0$.
QIV.A.1 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, the spectral radius $\rho(A)$ is a dominant eigenvalue of $A$ and the associated eigenspace is a line that possesses a strictly positive direction vector $x > 0$. We denote $r$ the spectral radius of $A$. We denote $x$ (respectively $y$) a strictly positive direction vector of the line $D = \operatorname{Ker}(A - rI_n)$ (respectively of the line $\Delta = \operatorname{Ker}(A^\top - rI_n)$). We denote $H = \operatorname{Im}(A - rI_n)$.
Show that $H$ is the hyperplane orthogonal to the line $\Delta$ (that is $H = \Delta^\perp$).
QIV.A.2 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, the spectral radius $\rho(A)$ is a dominant eigenvalue of $A$ and the associated eigenspace is a line that possesses a strictly positive direction vector $x > 0$. We denote $r$ the spectral radius of $A$. We denote $x$ (respectively $y$) a strictly positive direction vector of the line $D = \operatorname{Ker}(A - rI_n)$ (respectively of the line $\Delta = \operatorname{Ker}(A^\top - rI_n)$). We denote $H = \operatorname{Im}(A - rI_n)$. If necessary by multiplying $y$ by an appropriate strictly positive coefficient, we assume $(y \mid x) = y^\top x = 1$. We denote $L = xy^\top$.
Prove that $L$ is the matrix, in the canonical basis, of the projection of $\mathbb{R}^n$ onto the line $D$, parallel to the hyperplane $H$.
QIV.A.3 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, the spectral radius $\rho(A)$ is a dominant eigenvalue of $A$ and the associated eigenspace is a line that possesses a strictly positive direction vector $x > 0$. We denote $r$ the spectral radius of $A$. We denote $x$ (respectively $y$) a strictly positive direction vector of the line $D = \operatorname{Ker}(A - rI_n)$ (respectively of the line $\Delta = \operatorname{Ker}(A^\top - rI_n)$). We denote $H = \operatorname{Im}(A - rI_n)$. If necessary by multiplying $y$ by an appropriate strictly positive coefficient, we assume $(y \mid x) = y^\top x = 1$. We denote $L = xy^\top$.
Verify that $L$ has rank 1, that it is strictly positive, and that $L^\top y = y$.
QIV.A.4 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, the spectral radius $\rho(A)$ is a dominant eigenvalue of $A$ and the associated eigenspace is a line that possesses a strictly positive direction vector $x > 0$. We denote $r$ the spectral radius of $A$. We denote $x$ (respectively $y$) a strictly positive direction vector of the line $D = \operatorname{Ker}(A - rI_n)$ (respectively of the line $\Delta = \operatorname{Ker}(A^\top - rI_n)$). We denote $H = \operatorname{Im}(A - rI_n)$. If necessary by multiplying $y$ by an appropriate strictly positive coefficient, we assume $(y \mid x) = y^\top x = 1$. We denote $L = xy^\top$.
Show that $AL = LA = rL$. Deduce: $\forall m \in \mathbb{N}^*, (A - rL)^m = A^m - r^m L$.
QIV.B.1 Roots of polynomials Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$.
Let $\lambda$ be a nonzero eigenvalue of $B$ and let $z$ be an associated eigenvector.
Show that $Lz = 0$, then $Az = \lambda z$. Deduce $\rho(B) \leqslant r$.
QIV.B.2 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$.
Let $\lambda$ be a nonzero eigenvalue of $B$ and let $z$ be an associated eigenvector. By contradiction, we assume $\rho(B) = r$. We can therefore choose $\lambda$ such that $|\lambda| = r$. Show that then $\lambda = r$ then $Lz = z$ and reach a contradiction. Conclude.
QIV.B.3 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$, $L = xy^\top$ where $x > 0$ is a direction vector of $\operatorname{Ker}(A - rI_n)$ and $y > 0$ is a direction vector of $\operatorname{Ker}(A^\top - rI_n)$ with $y^\top x = 1$. We set $B = A - rL$, and we have shown $\rho(B) < r$ and $\forall m \in \mathbb{N}^*, (A - rL)^m = A^m - r^m L$.
Deduce from the above (and from subsection IV.A) that $\lim_{m \rightarrow +\infty} \left(\frac{1}{r}A\right)^m = L$.
QIV.C Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$. Let $\mu$ be the multiplicity of $r$ as an eigenvalue of $A$ and let $T = PAP^{-1}$ be a triangular reduction of $A$.
By examining the diagonal of $\left(\frac{1}{r}T\right)^m$ when $m \rightarrow +\infty$, show that $\mu = 1$.
QV.A.1 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$. We say that $A$ is irreducible if, for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \geqslant 0$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.
Express the irreducibility of $A$ in terms of paths in $A$.
QV.A.2 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$, irreducible.
Show that if $A$ is irreducible, then for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \in \llbracket 0, n-1 \rrbracket$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.
QV.A.3 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
Give a simple example of a square irreducible matrix that is not primitive.
QV.A.4 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$.
Show that if $A$ is not irreducible, then $A^2$ is not irreducible.
On the other hand, give a simple example of an irreducible matrix $A$ such that $A^2$ is not irreducible.
QV.A.5 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Show that the spectral radius of an irreducible matrix is strictly positive.
QV.B.1 Invariant lines and eigenvalues and vectors Matrix Power Computation and Application View
For the positive matrix $A$ of $\mathcal{M}_n(\mathbb{R})$, show that the following conditions are equivalent:
  • the matrix $A$ is irreducible;
  • the matrix $B = I_n + A + A^2 + \cdots + A^{n-1}$ is strictly positive;
  • the matrix $C = (I_n + A)^{n-1}$ is strictly positive.
QV.B.2 Invariant lines and eigenvalues and vectors Structured Matrix Characterization View
Let $A$ be irreducible. Show that no row (and no column) of $A$ is identically zero.
QV.C.1 Invariant lines and eigenvalues and vectors Matrix Power Computation and Application View
In this question, $A$ is a given irreducible matrix.
Suppose that $\forall i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A^{n-1} > 0$ (so $A$ is primitive). Reason in terms of paths in $A$.
QV.C.2 Invariant lines and eigenvalues and vectors Matrix Power Computation and Application View
In this question, $A$ is a given irreducible matrix.
Suppose that: $\exists i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A$ is primitive.
For all $j$ and $k$ in $\llbracket 1, n \rrbracket$, one can show that there exists in $A$ a path from $j$ to $k$ passing through $i$, and consider the maximum $m$ of the lengths of the paths thus obtained. One will prove that $A^m > 0$.
QVI.A Invariant lines and eigenvalues and vectors Matrix Power Computation and Application View
Let $A$ be an imprimitiv matrix with coefficient of imprimitivity $p \geqslant 2$.
For any integer $m$ that is not a multiple of $p$, show that the diagonal of $A^m$ is identically zero. One can be interested in the trace of $A^m$.
Deduce that the result of question IV.B.3 no longer holds if $A$ is imprimitiv.
QVI.B.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $Z_n$ is $X(X^{n-1} - 2)$.
Deduce that $Z_n$ is imprimitiv and specify its coefficient of imprimitivity.
QVI.B.3 Matrices Matrix Power Computation and Application View
We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that $Z_n^{n^2-2n+2} = 2^{n-1} Z_n$ and recover the fact that $Z_n$ is not primitive.
QVI.C.1 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$ (reminder: by convention, $p = 1$ if $A$ is primitive). Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$.
We recall that the spectrum of $A$ is invariant under the map $z \mapsto \omega z$, where $\omega = \exp(2\mathrm{i}\pi/p)$.
Deduce that, for all $k \in \{k_1, k_2, \ldots, k_s\}$, the integer $k$ is divisible by $p$. Think of the elementary symmetric functions of the $\lambda_i$.
QVI.C.2 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$.
Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.
QVI.D Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathcal{M}_n(\mathbb{R})$ be an irreducible matrix. For all $i$ in $\llbracket 1, n \rrbracket$, we denote $L_i = \{m \in \mathbb{N}^*, a_{i,i}^{(m)} > 0\}$ the (nonempty) set of lengths of circuits of $A$ passing through $i$, and we denote $d_i$ the gcd of the elements of $L_i$.
Establish that the coefficient of imprimitivity $p$ of $A$ is equal to $d_i$ for all $i$ in $\llbracket 1, n \rrbracket$ (this gcd does not depend on the index $i$).