grandes-ecoles

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. (The inner product on $\mathcal{M}_{n}(\mathbb{R})$ is given by $(M,N) \mapsto \operatorname{tr}(M^{\top}N)$.)

QI.A.2 Matrices Projection and Orthogonality View

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R})$, $\|A - A_{s}\|_{2} \leqslant \|A - S\|_{2}$. 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 $X^{\top}MY$ 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 $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}) \setminus \{0\}, X^{\top}A_{s}X > 0$.

QI.B.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}(A_{s}) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}(A_{s})$.
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 consider $A \in \mathcal{M}_{n}(\mathbb{R})$ and 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 $\operatorname{det}(A) = \operatorname{det}(A_{s})\operatorname{det}(I_{n} + Q)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}(A_{s})$.

QI.B.4 Matrices Determinant and Rank Computation View

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. We assume $A$ is invertible and, in accordance with the notations of the problem, $(A^{-1})_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2}\operatorname{det}\left((A^{-1})_{s}\right) = \operatorname{det}(A_{s})$.
One may consider $A(A^{-1})_{s}A^{\top}$.

QI.C.1 Matrices Eigenvalue and Characteristic Polynomial Analysis 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 $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for which there does not exist a matrix $A \in \mathrm{O}_{2}(\mathbb{R})$ satisfying $A_{s} = S$.

QI.C.3 Matrices Diagonalizability and Similarity 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 $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ 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

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix in $\mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$.
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 denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. 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 denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. 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 $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0 \end{pmatrix} \in \mathcal{M}_{n+1}(\mathbb{R})$ is singular.

QII.A.4 3x3 Matrices Block Matrix Multiplication and Determinant Identity View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 2$. 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 $B = \begin{pmatrix} B_{1} & B_{2} \\ B_{3} & B_{4} \end{pmatrix}$ 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: $$A_{N}B = \begin{pmatrix} I_{n} & 0 \\ N^{\top}A^{-1} & -N^{\top}A^{-1}N \end{pmatrix}$$

QII.A.5 3x3 Matrices Block Matrix Multiplication and Determinant Identity View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 2$. 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 $\mathcal{M}_{n}(\mathbb{R})$, and $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0 \end{pmatrix}$.
Deduce that $\operatorname{det}(A_{N}) = -N^{\top}A^{-1}N\operatorname{det}(A)$.

QII.A.6 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that if $\operatorname{det}\left((A^{-1})_{s}\right) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.

QII.A.7 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that if $\operatorname{det}(A_{s}) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.

QII.A.8 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. $A$ 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 3x3 Matrices Determinant of Parametric or Structured Matrix View

We consider the matrix $$A = A(\mu) = \begin{pmatrix} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{pmatrix}$$
Show that $A(\mu)$ is invertible for every real $\mu$.

QII.B.2 3x3 Matrices Determinant of Parametric or Structured Matrix View

We consider the matrix $$A = A(\mu) = \begin{pmatrix} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{pmatrix}$$
Calculate $A(\mu)_{s}$ and show that $A(\mu)_{s}$ is singular for $\mu = 1, 1-\sqrt{3}, 1+\sqrt{3}$.

QII.B.3 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We consider the matrix $$A = A(\mu) = \begin{pmatrix} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{pmatrix}$$
A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$.
Determine a hyperplane $H$ such that $A(1)$ is $H$-singular.

QII.C.1 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 3$. 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 $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$.
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 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 3$. 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 $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$.
Deduce that $A$ is $F$-singular if and only if the matrix $$A_{N} = \begin{pmatrix} A & N_{1} & N_{2} \\ N_{1}^{\top} & 0 & 0 \\ N_{2}^{\top} & 0 & 0 \end{pmatrix} = \begin{pmatrix} A & N \\ N^{\top} & 0_{2} \end{pmatrix} \in \mathcal{M}_{n+2}(\mathbb{R})$$ is singular.

QII.C.3 3x3 Matrices Block Matrix Multiplication and Determinant Identity View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. 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 $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists a matrix $B = \begin{pmatrix} B_{1} & B_{2} \\ B_{3} & B_{4} \end{pmatrix}$ 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 $$A_{N}B = \begin{pmatrix} I_{n} & 0 \\ N^{\top}A^{-1} & -N^{\top}A^{-1}N \end{pmatrix}$$

QII.C.4 3x3 Matrices Block Matrix Multiplication and Determinant Identity View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. 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 $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$, and $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0_{2} \end{pmatrix}$.
Deduce that $\operatorname{det}(A_{N}) = \operatorname{det}(N^{\top}A^{-1}N)\operatorname{det}(A)$.

QII.C.5 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$.
Show that there exists $P \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}(P^{\top}A^{-1}P) = 0$ if and only if there exists $P' \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}(P'^{\top}AP') = 0$.

QII.C.6 3x3 Matrices Block Matrix Multiplication and Determinant Identity View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Let $N' = \begin{pmatrix} N_{1}' & N_{2}' \end{pmatrix}$.
Show that $$\operatorname{det}(N'^{\top}AN') = \left(N_{1}'^{\top}A_{s}N_{1}'\right)\left(N_{2}'^{\top}A_{s}N_{2}'\right) - \left(N_{1}'^{\top}A_{s}N_{2}'\right)^{2} + \left(N_{1}'^{\top}A_{a}N_{2}'\right)^{2}$$

QII.C.7 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. 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 $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $\operatorname{det}(N^{\top}A^{-1}N) > 0$.

QII.C.8 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 3$. $A$ 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 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We return to the example of subsection II.B with $\mu = 1$, i.e. $$A(1) = \begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$.
How should we choose $N' = \begin{pmatrix} N_{1}' & N_{2}' \end{pmatrix}$ so that $\operatorname{det}(N'^{\top}A(1)N') = 0$?

QII.D.2 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We return to the example of subsection II.B with $\mu = 1$, i.e. $$A(1) = \begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$.
Determine a vector subspace $F$ of $E_{3}$ such that $\dim F = 1$ and such that $A(1)$ is $F$-singular.

QII.E.1 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$. 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 $\operatorname{det}(N'^{\top}AN') = 0$ for a matrix $N' \in \mathcal{G}_{n,p}(\mathbb{R})$ that one will define.

QII.E.2 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. Let $N' \in \mathcal{G}_{n,p}(\mathbb{R})$ be a matrix whose columns form a basis of $F^{\perp}$. 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 $X^{\top}N'^{\top}AN'X > 0$.

QII.E.3 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. Let $N' \in \mathcal{G}_{n,p}(\mathbb{R})$ be a matrix whose columns form a basis of $F^{\perp}$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
Deduce that the real eigenvalues of $N'^{\top}AN'$ are strictly positive.

QII.E.4 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. Let $N' \in \mathcal{G}_{n,p}(\mathbb{R})$ be a matrix whose columns form a basis of $F^{\perp}$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
Deduce that $\operatorname{det}(N'^{\top}AN') > 0$.

QII.E.5 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We 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 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
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 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
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 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. 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 $\bar{X}^{\top}AX \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}(\bar{X}^{\top}AX) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ is positively stable.

QIII.A.4 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
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 $v = u' + \lambda u$ is bounded on $\mathbb{R}^{+}$. Show that $u$ is bounded on $\mathbb{R}^{+}$.
One may consider the differential equation $y' + \lambda y = v$.

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 $t \in \mathbb{R}^{+}$, $$U(t) = \begin{pmatrix} u_{1}(t) \\ \vdots \\ u_{n}(t) \end{pmatrix}$$
Suppose that, for all $t \in \mathbb{R}^{+}$, $U'(t) + TU(t) = 0$.
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

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Recall that for any matrix $M \in \mathcal{M}_{n}(\mathbb{C})$, $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$.
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 $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}(\lambda_{j})$.
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 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. 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 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. 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 $\forall M \in \mathcal{M}_{n}(\mathbb{R}), \Phi(M) = A^{\top}M + MA$.
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 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. Recall that $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$ for any $M \in \mathcal{M}_{n}(\mathbb{C})$. For all real $t$, we set $V(t) = \exp(-tA^{\top})\exp(-tA)$ 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.

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2017 centrale-maths1__mp