LFM Pure

grandes-ecoles 2018 Q6 Compute eigenvalues of a given matrix View

Let $V = {}^{ t } \left( v _ { 1 } , \ldots , v _ { n } \right)$ be an eigenvector of $A _ { n }$ associated with a complex eigenvalue $\lambda$, where $A_n$ is the square matrix of size $n$:
$$A _ { n } = \left( \begin{array} { c c c c c c } 2 & - 1 & 0 & \ldots & \ldots & 0 \\ - 1 & 2 & - 1 & \ddots & & \vdots \\ 0 & - 1 & 2 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 2 & - 1 \\ 0 & \ldots & \ldots & 0 & - 1 & 2 \end{array} \right)$$
Show that $\lambda$ is necessarily real and that the components $v _ { i }$ of $V$ satisfy the relation:
$$v _ { i + 1 } - ( 2 - \lambda ) v _ { i } + v _ { i - 1 } = 0, \quad 1 \leq i \leq n$$
where we set $v _ { 0 } = v _ { n + 1 } = 0$.

grandes-ecoles 2018 Q7 Eigenvalue constraints from matrix properties View

Show that every eigenvalue of $A _ { n }$ is in the interval $]0,4[$.

grandes-ecoles 2018 Q8 Compute eigenvalues of a given matrix View

Let $\lambda$ be an eigenvalue of $A _ { n }$.
(a) Show that the complex roots $r _ { 1 } , r _ { 2 }$ of the polynomial
$$P ( r ) = r ^ { 2 } - ( 2 - \lambda ) r + 1$$
are distinct and conjugate.
(b) We set $r _ { 1 } = \overline { r _ { 2 } } = \rho e ^ { i \theta }$ with $\rho > 0$ and $\theta \in \mathbb { R }$.
Show that we necessarily have $\sin ( ( n + 1 ) \theta ) = 0$ and $\rho = 1$.

grandes-ecoles 2018 Q9 Spectral properties of structured or special matrices View

Determine the set of eigenvalues of $A _ { n }$ and a basis of eigenvectors.

grandes-ecoles 2018 Q10 Roots of Unity and Cyclotomic Properties View

Show that $r_1$ and $r_2$ are nonzero and that $r_1/r_2$ belongs to $\mathbb{U}_{n+1}$.

grandes-ecoles 2018 Q11 Roots of Unity and Cyclotomic Properties View

Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that $$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$

grandes-ecoles 2018 Q12 Roots of Unity and Cyclotomic Properties View

Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.

grandes-ecoles 2018 Q13 Diagonalize a matrix explicitly View

Conclude that $A_n(a,b,c)$ is diagonalizable and give its eigenvalues.

grandes-ecoles 2018 Q14 Matrix Power Computation and Application View

We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Calculate $M_n^2, \ldots, M_n^n$. Show that $M_n$ is invertible and give an annihilating polynomial of $M_n$.

grandes-ecoles 2018 Q15 Diagonalize a matrix explicitly View

We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Justify that $M_n$ is diagonalizable. Specify its eigenvalues (expressed using $\omega_n$) and give a basis of eigenvectors of $M_n$.

grandes-ecoles 2018 Q16 Diagonalizability and Similarity View

We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
We set $\Phi_n = \left(\omega_n^{(p-1)(q-1)}\right)_{1 \leqslant p,q \leqslant n} \in \mathcal{M}_n(\mathbb{C})$. Justify that $\Phi_n$ is invertible and give without calculation the value of the matrix $\Phi_n^{-1} M_n \Phi_n$.

grandes-ecoles 2018 Q18 Structured Matrix Characterization View

Conversely, if $P \in \mathbb{C}[X]$, show, using a Euclidean division of $P$ by a suitably chosen polynomial, that $P(M_n)$ is a circulant matrix.

grandes-ecoles 2018 Q19 Structured Matrix Characterization View

Show that the set of circulant matrices is a vector subspace of $\operatorname{Toep}_n(\mathbb{C})$, stable under multiplication and transposition.

grandes-ecoles 2018 Q20 Spectral properties of structured or special matrices View

Show that every circulant matrix is diagonalizable. Specify its eigenvalues and a basis of eigenvectors.

grandes-ecoles 2018 Q21 Linear Transformation and Endomorphism Properties View

Show that if $M$ is in $\mathcal{M}_n(\mathbb{C})$, then the following propositions are equivalent:
i. there exists $x_0$ in $\mathbb{C}^n$ such that $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ is a basis of $\mathbb{C}^n$;
ii. $M$ is similar to the matrix $C(a_0, \ldots, a_{n-1})$ defined by $$C(a_0, \ldots, a_{n-1}) = \left(\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_0 \\ 1 & \ddots & & \vdots & a_1 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & a_{n-1} \end{array}\right)$$ where $(a_0, \ldots, a_{n-1})$ are complex numbers.

grandes-ecoles 2018 Q22 Diagonalizability determination or proof View

Let $M$ be in $\mathcal{M}_n(\mathbb{C})$. We assume that $f_M$ is diagonalizable. We denote by $(\lambda_1, \ldots, \lambda_n)$ its eigenvalues (not necessarily distinct) and by $(e_1, \ldots, e_n)$ a basis of eigenvectors associated with these eigenvalues. Let $u = \sum_{i=1}^{n} u_i e_i$ be a vector of $\mathbb{C}^n$ where $(u_1, \ldots, u_n)$ are $n$ complex numbers.
Give a necessary and sufficient condition on $(u_1, \ldots, u_n, \lambda_1, \ldots, \lambda_n)$ for $(u, f_M(u), \ldots, f_M^{n-1}(u))$ to be a basis of $\mathbb{C}^n$.

grandes-ecoles 2018 Q23 Diagonalizability determination or proof View

Deduce a necessary and sufficient condition for a diagonalizable endomorphism to be cyclic. Then characterize its cyclic vectors.

grandes-ecoles 2018 Q26 Diagonalizability and Similarity View

Deduce a necessary and sufficient condition for a cyclic matrix to be diagonalizable.

grandes-ecoles 2018 Q27 Linear Transformation and Endomorphism Properties View

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $P \in \mathbb{C}[X]$. Show that $P(f_M) \in \mathcal{C}(f_M)$, where $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$.

grandes-ecoles 2018 Q28 Linear Transformation and Endomorphism Properties View

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $g \in \mathcal{C}(f_M)$. Show that there exist $(\alpha_0, \ldots, \alpha_{n-1}) \in \mathbb{C}^n$ such that $g = \alpha_0 Id_{\mathbb{C}^n} + \alpha_1 f_M + \cdots + \alpha_{n-1} f_M^{n-1}$. One may use the basis $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ and express $g(x_0)$ in this basis.

grandes-ecoles 2018 Q29 Linear Transformation and Endomorphism Properties View

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. The set $\mathcal{C}(f_M) = \{g \in \mathcal{L}(\mathbb{C}^n) \mid f_M \circ g = g \circ f_M\}$ is sought to be shown to be the set of polynomials in $f_M$. Conclude.

grandes-ecoles 2019 Q7 Automorphism and Endomorphism Structure View

Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.

grandes-ecoles 2019 Q8 Decomposition and Basis Construction View

Let $x$ be a non-zero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$

grandes-ecoles 2019 Q9 Subgroup and Normal Subgroup Properties View

Justify that $\operatorname{Vect}\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is stable under $f$.

grandes-ecoles 2019 Q9 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 $\mathcal{M}_{2}(\mathbb{C})$.

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

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Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

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Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

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Chain Rule 281

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Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164