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grandes-ecoles 2019 Q18 Diagonalizability and Similarity View
In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Is the matrix $J$ diagonalisable in $\mathcal{M}_{n}(\mathbb{R})$?
grandes-ecoles 2019 Q18 Matrix Norm, Convergence, and Inequality View
We denote by $\lambda _ { 1 }$ (respectively $\lambda _ { N }$) the smallest (respectively largest) eigenvalue of $A$, and we define $$\Lambda _ { k } = \{ Q \in \mathbb { R } [ X ] \mid \operatorname { deg } ( Q ) \leq k , Q ( 0 ) = 1 \}$$
Show that $$\left\| e _ { k } \right\| _ { A } \leq \left\| e _ { 0 } \right\| _ { A } \min _ { Q \in \Lambda _ { k } } \max _ { t \in \left[ \lambda _ { 1 } , \lambda _ { N } \right] } | Q ( t ) |$$
grandes-ecoles 2019 Q19 Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and that $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$ where $(u_1, \ldots, u_n)$ is the basis described in Q17. Justify that $f$ is cyclic.
grandes-ecoles 2019 Q19 Linear Transformation and Endomorphism Properties View
We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. There exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with Jordan blocks of sizes $m_k$ associated to eigenvalues $\lambda_k$. We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Justify that $f$ is cyclic.
grandes-ecoles 2019 Q19 Linear System and Inverse Existence View
Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Assume that $P$ is an annihilating polynomial of $A$ nilpotent.
We denote by $m$ the multiplicity of 0 in $P$, which allows us to write $P = X^m Q$ where $Q$ is a polynomial in $\mathbb{C}[X]$ such that $Q(0) \neq 0$. Prove that $Q(A)$ is invertible and then that $P$ is a multiple of $X^p$ in $\mathbb{C}[X]$.
grandes-ecoles 2019 Q19 Eigenvalue and Characteristic Polynomial Analysis View
In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Determine the complex eigenvalues of $J$ and the associated eigenspaces.
grandes-ecoles 2019 Q20 Matrix Algebra and Product Properties View
We call the commutant of $f$ the set $\mathcal{C}(f) = \{g \in \mathcal{L}(E) \mid f \circ g = g \circ f\}$. Show that $\mathcal{C}(f)$ is a subalgebra of $\mathcal{L}(E)$.
grandes-ecoles 2019 Q20 Matrix Algebra and Product Properties View
We call the commutant of $f$ the set $\mathcal{C}(f) = \{g \in \mathcal{L}(E) \mid f \circ g = g \circ f\}$. Show that $\mathcal{C}(f)$ is a subalgebra of $\mathcal{L}(E)$.
grandes-ecoles 2019 Q20 Determinant and Rank Computation View
We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Calculate the trace and rank of $A$. Deduce, without any calculation, the characteristic polynomial of $A$. Show that $A$ is nilpotent and give its nilpotency index.
grandes-ecoles 2019 Q20 Diagonalizability and Similarity View
In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Is the subset $\mathcal{A}$ a subalgebra of $\mathcal{M}_{n}(\mathbb{C})$?
grandes-ecoles 2019 Q21 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$. Justify the existence of $\lambda_0, \lambda_1, \ldots, \lambda_{n-1}$ of $\mathbb{K}$ such that
$$g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$$
grandes-ecoles 2019 Q21 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$.
Justify the existence of $\lambda_0, \lambda_1, \ldots, \lambda_{n-1}$ of $\mathbb{K}$ such that
$$g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$$
grandes-ecoles 2019 Q21 Diagonalizability and Similarity View
We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Prove that $A$ is similar to the matrix $\operatorname{diag}\left(J_2, J_1\right)$. Give the value of an invertible matrix $P$ such that $A = P \operatorname{diag}\left(J_2, J_1\right) P^{-1}$.
grandes-ecoles 2019 Q21 Diagonalizability and Similarity View
In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Show that there exists $P \in \mathrm{GL}_{n}(\mathbb{C})$ such that, for every matrix $A \in \mathcal{A}$, the matrix $P^{-1}AP$ is diagonal.
grandes-ecoles 2019 Q22 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$ with $g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$. Show then that $g \in \mathbb{K}[f]$.
grandes-ecoles 2019 Q22 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$, and suppose $g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$.
Show then that $g \in \mathbb{K}[f]$.
grandes-ecoles 2019 Q22 Linear Transformation and Endomorphism Properties View
We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Prove that $\operatorname{Im} u$ and $\operatorname{Ker} u$ are stable under $\rho$ and that $\rho$ is nilpotent.
grandes-ecoles 2019 Q22 Eigenvalue and Characteristic Polynomial Analysis View
In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. We denote by $Q \in \mathbb{R}[X]$ the polynomial $\sum_{k=0}^{n-1} a_{k} X^{k}$.
What are the complex eigenvalues of the matrix $J(a_{0}, \ldots, a_{n-1})$?
grandes-ecoles 2019 Q23 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic. Establish that $g \in \mathcal{C}(f)$ if and only if there exists a polynomial $R \in \mathbb{K}_{n-1}[X]$ such that $g = R(f)$.
grandes-ecoles 2019 Q23 Linear Transformation and Endomorphism Properties View
We assume that $f$ is cyclic. Establish that $g \in \mathcal{C}(f)$ if and only if there exists a polynomial $R \in \mathbb{K}_{n-1}[X]$ such that $g = R(f)$.
grandes-ecoles 2019 Q23 Linear Transformation and Endomorphism Properties View
We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Deduce the set of square roots of $A$. One may consider $R' = P^{-1}RP$.
grandes-ecoles 2019 Q23 Matrix Norm, Convergence, and Inequality View
Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. The trace of any matrix $M$ of $\mathcal{M}_{n}(\mathbb{R})$ is denoted $\operatorname{tr}(M)$.
Show that the map defined on $\mathcal{M}_{n}(\mathbb{R}) \times \mathcal{M}_{n}(\mathbb{R})$ by $(A, B) \mapsto \langle A \mid B \rangle = \operatorname{tr}(A^{\top} B)$ is an inner product on $\mathcal{M}_{n}(\mathbb{R})$.
grandes-ecoles 2019 Q23 Matrix Norm, Convergence, and Inequality View
We denote by $\kappa = \lambda _ { N } / \lambda _ { 1 }$. Show that the real number $\alpha$ from question 22 equals $\alpha = \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 }$ and deduce that $$\left\| e _ { k } \right\| _ { A } \leq 2 \left\| e _ { 0 } \right\| _ { A } \left( \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 } \right) ^ { k }$$
grandes-ecoles 2019 Q24 Matrix Power Computation and Application View
We propose to study the matrix equation $R^2 = J_3$.
Let $R$ be a solution of this equation. Give the values of $R^4$ and $R^6$, then the set of solutions of the equation.
grandes-ecoles 2019 Q24 Projection and Orthogonality View
Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ (with respect to the inner product $(A,B) \mapsto \operatorname{tr}(A^\top B)$) and we denote by $r$ its dimension.
What relationship holds between $d$ and $r$?

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