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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
2017 centrale-maths1__official

45 maths questions

QI.A.1 Matrices Projection and Orthogonality View
Show that $\mathcal{S}_{n}(\mathbb{R})$ and $\mathcal{A}_{n}(\mathbb{R})$ are two supplementary orthogonal vector subspaces in $\mathcal{M}_{n}(\mathbb{R})$ and specify their dimensions.
QI.A.2 Matrices Projection and Orthogonality View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R}), \left\|A - A_{s}\right\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.
QI.B.1 Matrices Projection and Orthogonality View
If $M \in \mathcal{M}_{n}(\mathbb{R})$ and $X, Y \in \mathcal{M}_{n,1}(\mathbb{R})$, the matrix $X^{\top} M Y$ belongs to $\mathcal{M}_{1}(\mathbb{R})$ and we agree to identify it with the real number equal to its unique entry.
With this convention, show that $A_{s} \in \mathcal{S}_{n}^{+}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}), X^{\top} A_{s} X \geqslant 0$ and that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}) \backslash \{0\}, X^{\top} A_{s} X > 0$.
QI.B.2 Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities View
For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right)$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ then $A$ is invertible.
QI.B.3 Matrices Matrix Decomposition and Factorization View
We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.
b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}\left(A_{s}\right) \operatorname{det}\left(I_{n} + Q\right)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}\left(A_{s}\right)$.
QI.B.4 Matrices Determinant and Rank Computation View
We assume $A$ is invertible and, in accordance with the notation of the problem, $\left(A^{-1}\right)_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2} \operatorname{det}\left(\left(A^{-1}\right)_{s}\right) = \operatorname{det}\left(A_{s}\right)$.
One may consider $A\left(A^{-1}\right)_{s} A^{\top}$.
QI.C.1 Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities View
Let $A \in \mathrm{O}_{n}(\mathbb{R})$. Show that the eigenvalues of $A_{s}$ are in $[-1,1]$.
QI.C.2 Matrices Structured Matrix Characterization View
Give an example of a symmetric matrix $S$ in $\mathcal{S}_{2}(\mathbb{R})$ such that $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for which there does not exist a matrix $A \in \mathrm{O}_{2}(\mathbb{R})$ satisfying $A_{s} = S$.
QI.C.3 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.
QII.A.1 Matrices Linear System and Inverse Existence View
Show that a matrix in $\mathcal{M}_{n}(\mathbb{R})$ is singular if and only if it is $E_{n}$-singular.
QII.A.2 Matrices Linear Transformation and Endomorphism Properties View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. Show that $A$ is $H$-singular if and only if there exist a non-zero vector $X$ in $H$ and a real number $\lambda$ such that $AX = \lambda N$.
QII.A.3 Matrices Determinant and Rank Computation View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. Deduce that $A$ is $H$-singular if and only if the matrix $A_{N} = \left(\begin{array}{cc} A & N \\ N^{\top} & 0 \end{array}\right) \in \mathcal{M}_{n+1}(\mathbb{R})$ is singular.
QII.A.4 Matrices Matrix Algebra and Product Properties View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,1}(\mathbb{R}), B_{3} \in \mathcal{M}_{1,n}(\mathbb{R})$, $B_{4} \in \mathcal{M}_{1}(\mathbb{R})$ such that: $A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$.
QII.A.5 Matrices Determinant and Rank Computation View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Deduce that $\operatorname{det}\left(A_{N}\right) = -N^{\top} A^{-1} N \operatorname{det}(A)$.
QII.A.6 Matrices Determinant and Rank Computation View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that if $\operatorname{det}\left(\left(A^{-1}\right)_{s}\right) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.
QII.A.7 Matrices Determinant and Rank Computation View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Deduce that if $\operatorname{det}\left(A_{s}\right) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.
QII.A.8 Matrices Projection and Orthogonality View
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that $A$ is $H$-regular for every hyperplane $H$ of $E_{n}$.
QII.B.1 Matrices Determinant and Rank Computation View
We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Show that $A(\mu)$ is invertible for every real $\mu$.
QII.B.2 Matrices Determinant and Rank Computation View
We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Calculate $A(\mu)_{s}$ and show that $A(\mu)_{s}$ is singular for $\mu = 1, 1-\sqrt{3}, 1+\sqrt{3}$.
QII.B.3 Matrices Projection and Orthogonality View
We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Determine a hyperplane $H$ such that $A(1)$ is $H$-singular.
QII.C.1 Matrices Linear Transformation and Endomorphism Properties View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$.
Show that $A$ is $F$-singular if and only if there exist a non-zero element $X$ of $F$ and two real numbers $\lambda_{1}$, $\lambda_{2}$ such that $AX = \lambda_{1} N_{1} + \lambda_{2} N_{2}$.
QII.C.2 Matrices Determinant and Rank Computation View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$.
Deduce that $A$ is $F$-singular if and only if the matrix $$A_{N} = \left(\begin{array}{ccc} A & N_{1} & N_{2} \\ N_{1}^{\top} & 0 & 0 \\ N_{2}^{\top} & 0 & 0 \end{array}\right) = \left(\begin{array}{cc} A & N \\ N^{\top} & 0_{2} \end{array}\right) \in \mathcal{M}_{n+2}(\mathbb{R})$$ is singular.
QII.C.3 Matrices Matrix Algebra and Product Properties View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,2}(\mathbb{R}), B_{3} \in \mathcal{M}_{2,n}(\mathbb{R})$ and $B_{4} \in \mathcal{M}_{2}(\mathbb{R})$ such that $$A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$$
QII.C.4 Matrices Determinant and Rank Computation View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that $\operatorname{det}\left(A_{N}\right) = \operatorname{det}\left(N^{\top} A^{-1} N\right) \operatorname{det}(A)$.
QII.C.5 Matrices Determinant and Rank Computation View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists $P \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\top} A^{-1} P\right) = 0$ if and only if there exists $P^{\prime} \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\prime\top} A P^{\prime}\right) = 0$.
QII.C.6 Matrices Determinant and Rank Computation View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that if $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ then $$\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = \left(N_{1}^{\prime\top} A_{s} N_{1}^{\prime}\right)\left(N_{2}^{\prime\top} A_{s} N_{2}^{\prime}\right) - \left(N_{1}^{\prime\top} A_{s} N_{2}^{\prime}\right)^{2} + \left(N_{1}^{\prime\top} A_{a} N_{2}^{\prime}\right)^{2}$$
QII.C.7 Matrices Matrix Algebra and Product Properties View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $\operatorname{det}\left(N^{\top} A^{-1} N\right) > 0$.
QII.C.8 Proof Deduction or Consequence from Prior Results View
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Conclude that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $A$ is $F$-regular for every vector subspace $F$ of dimension $n-2$ of $E_{n}$.
QII.D.1 Matrices Projection and Orthogonality View
We return to the example of subsection II.B with $\mu = 1$, i.e., $$A(1) = \left(\begin{array}{ccc} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)$$ How should we choose $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ so that $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = 0$?
QII.D.2 Matrices Linear Transformation and Endomorphism Properties View
We return to the example of subsection II.B with $\mu = 1$, i.e., $$A(1) = \left(\begin{array}{ccc} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)$$ Determine a vector subspace $F$ of $E_{3}$ such that $\dim F = 1$ and such that $A(1)$ is $F$-singular.
QII.E.1 Matrices Linear Transformation and Endomorphism Properties View
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. Show that $A$ is $F$-singular if $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = 0$ for a matrix $N^{\prime} \in \mathcal{G}_{n,p}(\mathbb{R})$ that one will define.
QII.E.2 Matrices Projection and Orthogonality View
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that if $X \in \mathcal{M}_{p,1}(\mathbb{R})$ is non-zero then $X^{\top} N^{\prime\top} A N^{\prime} X > 0$.
QII.E.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that the real eigenvalues of $N^{\prime\top} A N^{\prime}$ are strictly positive.
QII.E.4 Matrices Determinant and Rank Computation View
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) > 0$.
QII.E.5 Matrices Linear Transformation and Endomorphism Properties View
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that $A$ is $F$-regular for every non-zero vector subspace $F$ of $E_{n}$.
QIII.A.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathcal{M}_{2}(\mathbb{R})$. Show that $A$ is positively stable if and only if $\operatorname{tr}(A) > 0$ and $\operatorname{det}(A) > 0$.
QIII.A.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
a) Is the sum of two positively stable matrices of $\mathcal{M}_{2}(\mathbb{R})$ necessarily positively stable?
b) Let $A, B$ in $\mathcal{M}_{n}(\mathbb{R})$ be two positively stable matrices that commute. Show that $A + B$ is positively stable.
QIII.A.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ such that $A_{s}$ is positive definite.
a) Let $X = Y + \mathrm{i} Z$ be a column matrix of $\mathcal{M}_{n,1}(\mathbb{C})$, where $Y$ and $Z$ belong to $\mathcal{M}_{n,1}(\mathbb{R})$. We set $\bar{X} = Y - \mathrm{i} Z$ and we identify the matrix $\bar{X}^{\top} A X \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}\left(\bar{X}^{\top} A X\right) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ is positively stable.
QIII.A.4 Matrices Structured Matrix Characterization View
Give an example of a positively stable matrix $A$ such that $A_{s}$ is not positive definite.
QIII.B.1 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $\lambda \in \mathbb{C}$ such that $\operatorname{Re}(\lambda) > 0$. Let $u$ be a function with complex values of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$.
Suppose that the function $v = u^{\prime} + \lambda u$ is bounded on $\mathbb{R}^{+}$. Show that $u$ is bounded on $\mathbb{R}^{+}$.
One may consider the differential equation $y^{\prime} + \lambda y = v$.
QIII.B.2 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $T \in \mathcal{M}_{n}(\mathbb{C})$ be an upper triangular matrix with complex entries. Suppose that the diagonal entries of $T$ are complex numbers with strictly positive real part. Let $u_{1}, \ldots, u_{n}$ be functions with complex values, defined and of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$ and let, for all $t \in \mathbb{R}^{+}$, $$U(t) = \left(\begin{array}{c} u_{1}(t) \\ \vdots \\ u_{n}(t) \end{array}\right)$$ Suppose that, for all $t \in \mathbb{R}^{+}, U^{\prime}(t) + T U(t) = 0$.
Show that the functions $u_{j}$, where $1 \leqslant j \leqslant n$, are bounded on $\mathbb{R}^{+}$.
QIII.B.3 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix with complex eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ and let $\alpha$ be a real number such that $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}\left(\lambda_{j}\right)$.
Show that the function $t \mapsto \mathrm{e}^{\alpha t} \exp(-tA)$ is bounded on $\mathbb{R}^{+}$.
One may apply question III.B.2 to an upper triangular matrix $T$ similar to $A - \alpha I_{n}$.
QIII.C.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top} M + MA$$ Show that $\Phi$ is positively stable, that is, its matrix in any basis of $\mathcal{M}_{n}(\mathbb{R})$ is positively stable.
QIII.C.2 Matrices Linear Transformation and Endomorphism Properties View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top} M + MA$$
a) Show that there exists a unique matrix $B \in \mathcal{M}_{n}(\mathbb{R})$ such that $A^{\top} B + BA = I_{n}$.
b) Show that $B$ is symmetric and that $\operatorname{det}(B) > 0$.
QIII.C.3 Matrices Matrix Power Computation and Application View
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. For all real $t$, we set $V(t) = \exp\left(-tA^{\top}\right) \exp(-tA)$ and $W(t) = \int_{0}^{t} V(s) \mathrm{d}s$.
a) Show that, for all real $t, V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0, W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
b) Show that, for all real $t, A^{\top} W(t) + W(t) A = I_{n} - V(t)$.
c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.