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grandes-ecoles 2020 Q20 Diagonalizability and Similarity View
For all permutations $\rho, \rho' \in \mathfrak{S}_n$, show that $P_{\rho\rho'} = P_\rho P_{\rho'}$. Deduce that, for all permutations $\sigma, \tau \in \mathfrak{S}_n$, if $\sigma$ and $\tau$ are conjugate then $P_\sigma$ and $P_\tau$ are similar.
grandes-ecoles 2020 Q20 Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $A$ admits a strictly positive eigenvector associated with the eigenvalue 1.
grandes-ecoles 2020 Q20 Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $\lambda \in \mathrm{sp}_{\mathbb{R}}(M)$ and $p = \dim E_{\lambda}$. Let $(X_{1}, \ldots, X_{p})$ be a basis of $E_{\lambda}$. Show that $(J_{n} X_{1}, \ldots, J_{n} X_{p})$ is a basis of $E_{1/\lambda}$ and that $$\dim(E_{\lambda}) = \dim(E_{1/\lambda}).$$
grandes-ecoles 2020 Q21 Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that 1 is the only eigenvalue of $A$ with modulus 1.
One may admit without proof that if $z_1, z_2, \ldots, z_k$ are non-zero complex numbers such that $|z_1 + \cdots + z_k| = |z_1| + \cdots + |z_k|$, then $\forall j \in \llbracket 1, k \rrbracket, \exists \lambda_j \in \mathbb{R}^+$ such that $z_j = \lambda_j z_1$.
grandes-ecoles 2020 Q21 Projection and Orthogonality View
Let $Y_{1}, \ldots, Y_{p}$ be vectors of $\mathcal{M}_{2n,1}(\mathbb{R})$. Let $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$. Show the implication $$Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp} \Longrightarrow J_{n} Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, Y, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp}.$$
grandes-ecoles 2020 Q22 Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $\dim\left(\ker\left(A - I_n\right)\right) = 1$.
grandes-ecoles 2020 Q22 Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. In this question $\lambda = 1$. Show that $E_{1}$ has even dimension and that there exists a basis of $E_{1}$ that is orthonormal and of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$ where $2p$ is the dimension of $E_{1}$.
grandes-ecoles 2020 Q23 Eigenvalue and Characteristic Polynomial Analysis View
By combining the results of sub-parts II.B and II.C, justify that we have proved Proposition 1: If $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix, then $\rho(A)$ is a dominant eigenvalue of $A$. The associated eigenspace $\ker\left(A - \rho(A) I_n\right)$ is one-dimensional and is spanned by a strictly positive eigenvector.
grandes-ecoles 2020 Q23 Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. What about $E_{-1}$? (i.e., does $E_{-1}$ have even dimension and does there exist an orthonormal basis of $E_{-1}$ of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$?)
grandes-ecoles 2020 Q24 Determinant and Rank Computation View
Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$
grandes-ecoles 2020 Q24 Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $\lambda \in S = \operatorname{sp}(A) \setminus \{\rho(A)\}$. Let $Y \in \ker\left(A - \lambda I_n\right)$. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to 0.
grandes-ecoles 2020 Q24 Diagonalizability and Similarity View
Prove the following property: if $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.
grandes-ecoles 2020 Q24 Eigenvalue and Characteristic Polynomial Analysis View
Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$
grandes-ecoles 2020 Q25 Diagonalizability and Similarity View
Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.
grandes-ecoles 2020 Q25 Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $Y \in \mathcal{M}_{n,1}(\mathbb{R})$ be a positive vector. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to the projection of $Y$ onto $E_{\rho(A)}(A)$ parallel to $\bigoplus_{\lambda \in S} E_\lambda(A)$. Verify that, if it is non-zero, this latter vector (the projection of $Y$) is strictly positive.
grandes-ecoles 2020 Q25 Bilinear and Symplectic Form Properties View
Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Show that $A \in \mathcal{S}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R})$.
grandes-ecoles 2020 Q25 Diagonalizability and Similarity View
Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.
grandes-ecoles 2020 Q26 Eigenvalue and Characteristic Polynomial Analysis View
By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)
grandes-ecoles 2020 Q26 Diagonalizability and Similarity View
Justify that for all integer $k \geqslant 1$, $A^k$ is similar in $\mathcal{M}_n(\mathbb{C})$ to a triangular matrix, whose diagonal coefficients we will specify.
grandes-ecoles 2020 Q26 Matrix Decomposition and Factorization View
Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Construct an orthogonal and symplectic matrix $P$ such that $P^{\top} A P$ is diagonal.
grandes-ecoles 2020 Q26 Eigenvalue and Characteristic Polynomial Analysis View
By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)
grandes-ecoles 2020 Q27 Diagonalizability and Similarity View
Deduce property (S): The permutation matrices $P_\sigma$ and $P_\tau$ are similar if and only if the permutations $\sigma$ and $\tau$ are conjugate.
One may compute $T_\sigma D$ where $T_\sigma$ is the cycle type of $\sigma$ and $D$ is the divisor matrix defined in I.D.
grandes-ecoles 2020 Q27 Matrix Power Computation and Application View
Show that $\lim_{k \rightarrow +\infty} \frac{\operatorname{tr}\left(A^{k+1}\right)}{\operatorname{tr}\left(A^k\right)} = \rho(A)$.
grandes-ecoles 2020 Q27 Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $m$ be the linear map canonically associated with $M$. Show the equality $\mathrm{sp}_{\mathbb{R}}(M) = \emptyset$.
grandes-ecoles 2020 Q27 Diagonalizability and Similarity View
Deduce property (S): The permutation matrices $P_\sigma$ and $P_\tau$ are similar if and only if the permutations $\sigma$ and $\tau$ are conjugate.
One may compute $T_\sigma D$ where $T_\sigma$ is the cycle type of $\sigma$ and $D$ is the divisor matrix defined in I.D.

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