LFM Pure

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 Q25 Eigenvalue and Characteristic Polynomial Analysis View

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. If $\lambda$ is a root of the polynomial identified in Q24, determine the eigenspace of $C(a_0, \ldots, a_{n-1})$ associated with the eigenvalue $\lambda$ and specify its dimension.

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 System and Inverse Existence View

Let $\tau$ be a strictly positive real and $q$ a natural integer greater than or equal to 2. We set $\delta = \frac{1}{q+1}$ and $r = \frac{\tau}{\delta^{2}}$. The numerical scheme imposes, for any natural integer $n$ and any $k \in \llbracket 1, q \rrbracket$: $$\frac{f_{n+1}(k) - f_{n}(k)}{\tau} = \frac{f_{n}(k+1) - 2f_{n}(k) + f_{n}(k-1)}{\delta^{2}}$$ as well as $f_{n}(0) = f_{n}(q+1) = 0$. We set $F_{n} = \left(\begin{array}{c} f_{n}(1) \\ \vdots \\ f_{n}(q) \end{array}\right)$, $I_{q}$ is the identity matrix of order $q$, $B$ is the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise, and $A = (1-2r)I_{q} + rB$. Show that, for all $n \in \mathbb{N}$, $F_{n+1} = A F_{n}$.

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 Diagonalizability and Similarity View

With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), justify that the matrices $A$ and $B$ are diagonalizable over $\mathbb{R}$ and that, for all $n \in \mathbb{N}$, $F_{n} = A^{n} F_{0}$.

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 2018 Q31 Linear Transformation and Endomorphism Properties View

Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Is the matrix $N$ cyclic?

grandes-ecoles 2018 Q32 Structured Matrix Characterization View

Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Show that the set of matrices that commute with $N$ is the set of lower triangular Toeplitz matrices.

grandes-ecoles 2018 Q33 Matrix Algebra and Product Properties View

Show that if $i$ and $j$ are in $\llbracket -n+1, n-1 \rrbracket$, if $A \in \Delta_i$ and $B \in \Delta_j$, then $AB \in \Delta_{i+j}$.

grandes-ecoles 2018 Q34 Matrix Norm, Convergence, and Inequality View

With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.

grandes-ecoles 2018 Q34 Matrix Algebra and Product Properties View

Deduce that if $A \in H_i$ and $B \in H_j$, then $AB \in H_{i+j}$.

grandes-ecoles 2018 Q35 Matrix Norm, Convergence, and Inequality View

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a unit vector $u$ in $\mathbb{R}^{d}$ and define $g(M) = \|M \cdot u\|$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$\|M \cdot u\| \leqslant \|M\|_{F}$$

grandes-ecoles 2018 Q35 Matrix Norm, Convergence, and Inequality View

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$\|M \cdot u\| \leqslant \|M\|_{F}$$

grandes-ecoles 2018 Q35 Linear System and Inverse Existence View

Let $C$ be a nilpotent matrix. Show that $I_n + C$ is invertible and that $$\left(I_n + C\right)^{-1} = I_n - C + C^2 + \cdots + (-1)^{n-1} C^{n-1}$$

grandes-ecoles 2018 Q40 Linear Transformation and Endomorphism Properties View

We define the operators $$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$ Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.

grandes-ecoles 2018 Q41 Linear Transformation and Endomorphism Properties View

Show that $\mathcal{S}(\Delta_{k+1}) \subset \Delta_k$ and $\mathcal{S}^*(\Delta_k) \subset \Delta_{k+1}$.

grandes-ecoles 2018 Q42 Projection and Orthogonality View

We equip $\mathcal{M}_n(\mathbb{R})$ with its usual inner product defined by: $\forall (M_1, M_2) \in \mathcal{M}_n(\mathbb{R}), \langle M_1, M_2 \rangle = \operatorname{tr}({}^t M_1 M_2)$. We denote by $\mathcal{S}_{k+1}$ the restriction of $\mathcal{S}$ to $\Delta_{k+1}$ and $\mathcal{S}_k^*$ the restriction of $\mathcal{S}^*$ to $\Delta_k$.
Verify that for all $X$ in $\Delta_{k+1}$ and $Y$ in $\Delta_k$, $\langle \mathcal{S}_{k+1} X, Y \rangle = \langle X, \mathcal{S}_k^* Y \rangle$. Deduce that $\ker(\mathcal{S}_k^*)$ and $\operatorname{Im}(\mathcal{S}_{k+1})$ are orthogonal complements in $\Delta_k$, that is $$\Delta_k = \ker(\mathcal{S}_k^*) \oplus^{\perp} \operatorname{Im}(\mathcal{S}_{k+1})$$

grandes-ecoles 2018 Q43 Diagonalizability and Similarity View

Let $T$ be an upper triangular matrix, $A = N + T$ and $k \geqslant 0$. Show that $A$ is similar to a matrix $L$ whose diagonal coefficients of order $k$ are all equal and satisfying $\forall i \in \llbracket -1, k-1 \rrbracket, L^{(i)} = A^{(i)}$.

grandes-ecoles 2018 Q44 Diagonalizability and Similarity View

Deduce that every cyclic matrix is similar to a Toeplitz matrix.

grandes-ecoles 2019 Q1 Eigenvalue and Characteristic Polynomial Analysis View

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.

grandes-ecoles 2019 Q1 Eigenvalue and Characteristic Polynomial Analysis View

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.

grandes-ecoles 2019 Q1 Linear Transformation and Endomorphism Properties View

What can be said about a nilpotent endomorphism of index 1?

grandes-ecoles 2019 Q1 Matrix Norm, Convergence, and Inequality View

When $x \in \mathbb{C}^n$, verify that $\|x\|_2^2 = \bar{x}^T x$.

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144