grandes-ecoles

Q1 Matrices Structured Matrix Characterization View

Are the subsets $\mathrm{T}_{n}(\mathbb{K})$ and $\mathrm{T}_{n}^{+}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

Q2 Matrices Matrix Algebra and Product Properties View

Are the subsets $S_{2}(\mathbb{K})$ and $A_{2}(\mathbb{K})$ subalgebras of $\mathcal{M}_{2}(\mathbb{K})$?

Q3 Matrices Matrix Algebra and Product Properties View

Suppose $n \geqslant 3$. Are the subsets $\mathrm{S}_{n}(\mathbb{K})$ and $\mathrm{A}_{n}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

Q4 Matrices Linear Transformation and Endomorphism Properties View

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\mathcal{A}_{F}$ is a subalgebra of $\mathcal{L}(E)$.

Q5 Matrices Determinant and Rank Computation View

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\operatorname{dim} \mathcal{A}_{F} = n^{2} - pn + p^{2}$.
One may consider a basis of $E$ in which the matrix of every element of $\mathcal{A}_{F}$ is block triangular.

Q6 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Determine $\max_{1 \leqslant p \leqslant n-1} \left( n^{2} - pn + p^{2} \right)$.

Q7 Matrices Structured Matrix Characterization View

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\Gamma(\mathbb{K})$ is a subalgebra of $\mathcal{M}_{2}(\mathbb{K})$.

Q8 Matrices Diagonalizability and Similarity View

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\Gamma(\mathbb{R})$ is not a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{R})$.

Q9 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$ is diagonalisable over $\mathbb{C}$. Deduce that $\Gamma(\mathbb{C})$ is a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{C})$.

Q10 Matrices Matrix Power Computation and Application View

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
Specify the matrices $J$ and $J^{2}$. (One may distinguish the cases $n = 2$ and $n > 2$.)

Q11 Matrices Matrix Power Computation and Application View

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
Specify the matrices $J^{n}$ and $J^{k}$ for $2 \leqslant k \leqslant n-1$.

Q12 Matrices Structured Matrix Characterization View

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
What is the relationship between the matrix $J(a_{0}, \ldots, a_{n-1})$ and the $J^{k}$, where $0 \leqslant k \leqslant n-1$?

Q13 Matrices Structured Matrix Characterization 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
Show that $(I_{n}, J, J^{2}, \ldots, J^{n-1})$ is a basis of $\mathcal{A}$.

Q14 Matrices Structured Matrix Characterization 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
Let $M \in \mathcal{M}_{n}(\mathbb{R})$. Show that $M$ commutes with $J$ if and only if $M$ commutes with every element of $\mathcal{A}$.

Q15 Matrices Structured Matrix Characterization 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 $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. 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 $\mathbb{R}^{n}$.
Show that $\mathcal{A}$ is a commutative subalgebra of $\mathcal{M}_{n}(\mathbb{R})$.

Q16 Matrices 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 characteristic polynomial of $J$.

Q17 Matrices 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 $\mathbb{R}^{n}$.
Show that $J$ is diagonalisable in $\mathcal{M}_{n}(\mathbb{C})$.

Q18 Matrices 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 $\mathcal{M}_{n}(\mathbb{R})$?

Q19 Matrices 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.

Q20 Matrices 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 $\mathcal{M}_{n}(\mathbb{C})$?

Q21 Matrices 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.

Q22 Matrices 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 $J(a_{0}, \ldots, a_{n-1})$?

Q23 Matrices 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 $\mathcal{M}_{n}(\mathbb{R})$.

Q24 Matrices 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 $r$?

Q25 Matrices 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 $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Let $M \in \mathcal{M}_{n}(\mathbb{R})$. Show that $M$ belongs to $\mathcal{A}$ if and only if, for all $i \in \llbracket 1, r \rrbracket$, $\langle A_{i} \mid M \rangle = 0$.

Q26 Matrices 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 $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Show that for every matrix $N \in \mathcal{A}$ and all $i \in \llbracket 1, r \rrbracket$, we have $N^{\top} A_{i} \in \mathcal{A}^{\perp}$.

Q27 Matrices Structured Matrix Characterization 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. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$.
Show that $\mathcal{A}^{\top}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ of the same dimension as $\mathcal{A}$.

Q28 Matrices Linear Transformation and Endomorphism Properties 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 $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$. We denote by $\mathcal{M}_{n,1}(\mathbb{R})$ the $\mathbb{R}$-vector space of column matrices with $n$ rows and real coefficients.
Let $X \in \mathcal{M}_{n,1}(\mathbb{R})$ and let $F = \operatorname{Vect}(A_{1}X, \ldots, A_{r}X)$. Show that $F$ is stable by the endomorphisms of $\mathcal{M}_{n,1}(\mathbb{R})$ canonically associated with elements of $\mathcal{A}^{\top}$.

Q29 Matrices Determinant and Rank Computation 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 $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Show that $d \leqslant n^{2} - n + 1$ and conclude.

Q30 Matrices Linear Transformation and Endomorphism Properties View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We propose to prove by strong induction on $n \in \mathbb{N}^{*}$ that if all elements of $\mathcal{A}$ are nilpotent, then $\mathcal{A}$ is trigonalisable.
Show that the result is true if $n = 1$.

Q31 Matrices Linear Transformation and Endomorphism Properties View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result (that $\mathcal{A}$ is trigonalisable) is true for all natural integers $d \leqslant n-1$.
We admit Burnside's Theorem: Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geqslant 2$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$. If the only vector subspaces of $E$ stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$, then $\mathcal{A} = \mathcal{L}(E)$.
Show that there exists a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$.

Q32 Matrices Linear Transformation and Endomorphism Properties View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$.
Show that there exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.

Q33 Matrices Linear Transformation and Endomorphism Properties View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$. There exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.
Show that $\{ A(u) \mid u \in \mathcal{A} \}$ is a subalgebra of $\mathcal{M}_{r}(\mathbb{C})$ consisting of nilpotent matrices and that $\{ D(u) \mid u \in \mathcal{A} \}$ is a subalgebra of $\mathcal{M}_{s}(\mathbb{C})$ consisting of nilpotent matrices.

Q34 Matrices Diagonalizability and Similarity View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$. There exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.
Show that $\mathcal{A}$ is trigonalisable.

Q35 Matrices Diagonalizability and Similarity View

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms.
Show that there exists a basis of $E$ in which the matrices of elements of $\mathcal{A}$ belong to $\mathrm{T}_{n}^{+}(\mathbb{C})$.

Q36 Matrices Linear Transformation and Endomorphism Properties View

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Let $x$ and $y$ be two elements of $E$, with $x$ being non-zero. Show that there exists $u \in \mathcal{A}$ such that $u(x) = y$.
One may consider in $E$ the vector subspace $\{ u(x) \mid u \in \mathcal{A} \}$.

Q37 Matrices Determinant and Rank Computation View

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Let $v \in \mathcal{A}$ of rank greater than or equal to 2. Show that there exists $u \in \mathcal{A}$ and $\lambda \in \mathbb{C}$ such that $$0 < \operatorname{rg}(v \circ u \circ v - \lambda v) < \operatorname{rg} v.$$ Consider $x$ and $y$ in $E$ such that the family $(v(x), v(y))$ is free, justify the existence of $u \in \mathcal{A}$ such that $u \circ v(x) = y$ and consider the endomorphism induced by $v \circ u$ on $\operatorname{Im} v$.

Q38 Matrices Determinant and Rank Computation View

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$.

Q39 Matrices Linear Transformation and Endomorphism Properties View

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$). Let $u_{0} \in \mathcal{A}$ be of rank 1. We can therefore choose a basis $\mathcal{B} = (\varepsilon_{1}, \ldots, \varepsilon_{n})$ of $E$ such that $(\varepsilon_{2}, \ldots, \varepsilon_{n})$ is a basis of $\ker u_{0}$.
Show that there exist $u_{1}, \ldots, u_{n} \in \mathcal{A}$ of rank 1 such that $u_{i}(\varepsilon_{1}) = \varepsilon_{i}$ for all $i \in \llbracket 1, n \rrbracket$.

Q40 Matrices Linear Transformation and Endomorphism Properties View

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$). Let $u_{0} \in \mathcal{A}$ be of rank 1. We can therefore choose a basis $\mathcal{B} = (\varepsilon_{1}, \ldots, \varepsilon_{n})$ of $E$ such that $(\varepsilon_{2}, \ldots, \varepsilon_{n})$ is a basis of $\ker u_{0}$.
Conclude (that $\mathcal{A} = \mathcal{L}(E)$).

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2019 centrale-maths1__pc