LFM Pure

grandes-ecoles 2017 QI.C.2 Structured Matrix Characterization View

Give an example of a symmetric matrix $S$ in $\mathcal{S}_{2}(\mathbb{R})$ such that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for which there does not exist a matrix $A \in \mathrm{O}_{2}(\mathbb{R})$ satisfying $A_{s} = S$.

grandes-ecoles 2017 QI.C.3 Diagonalizability and Similarity View

Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.

grandes-ecoles 2017 QII.A.1 Linear System and Inverse Existence View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix in $\mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$.
Show that a matrix in $\mathcal{M}_{n}(\mathbb{R})$ is singular if and only if it is $E_{n}$-singular.

grandes-ecoles 2017 QII.A.2 Linear Transformation and Endomorphism Properties View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$.
Show that $A$ is $H$-singular if and only if there exist a non-zero vector $X$ in $H$ and a real number $\lambda$ such that $AX = \lambda N$.

grandes-ecoles 2017 QII.A.3 Determinant and Rank Computation View

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$.
Deduce that $A$ is $H$-singular if and only if the matrix $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0 \end{pmatrix} \in \mathcal{M}_{n+1}(\mathbb{R})$ is singular.

grandes-ecoles 2017 QI.A.1 Projection and Orthogonality View

Show that $\mathcal{S}_{n}(\mathbb{R})$ and $\mathcal{A}_{n}(\mathbb{R})$ are two supplementary orthogonal vector subspaces in $\mathcal{M}_{n}(\mathbb{R})$ and specify their dimensions.

grandes-ecoles 2017 QI.A.2 Projection and Orthogonality View

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R}), \left\|A - A_{s}\right\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.

grandes-ecoles 2017 QI.B.1 Projection and Orthogonality View

If $M \in \mathcal{M}_{n}(\mathbb{R})$ and $X, Y \in \mathcal{M}_{n,1}(\mathbb{R})$, the matrix $X^{\top} M Y$ belongs to $\mathcal{M}_{1}(\mathbb{R})$ and we agree to identify it with the real number equal to its unique entry.
With this convention, show that $A_{s} \in \mathcal{S}_{n}^{+}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}), X^{\top} A_{s} X \geqslant 0$ and that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ if and only if $\forall X \in \mathcal{M}_{n,1}(\mathbb{R}) \backslash \{0\}, X^{\top} A_{s} X > 0$.

grandes-ecoles 2017 QI.B.3 Matrix Decomposition and Factorization View

We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.
b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}\left(A_{s}\right) \operatorname{det}\left(I_{n} + Q\right)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}\left(A_{s}\right)$.

grandes-ecoles 2017 QI.B.4 Determinant and Rank Computation View

We assume $A$ is invertible and, in accordance with the notation of the problem, $\left(A^{-1}\right)_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2} \operatorname{det}\left(\left(A^{-1}\right)_{s}\right) = \operatorname{det}\left(A_{s}\right)$.
One may consider $A\left(A^{-1}\right)_{s} A^{\top}$.

grandes-ecoles 2017 QI.C.2 Structured Matrix Characterization View

Give an example of a symmetric matrix $S$ in $\mathcal{S}_{2}(\mathbb{R})$ such that $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for which there does not exist a matrix $A \in \mathrm{O}_{2}(\mathbb{R})$ satisfying $A_{s} = S$.

grandes-ecoles 2017 QII.A.1 Linear System and Inverse Existence View

Show that a matrix in $\mathcal{M}_{n}(\mathbb{R})$ is singular if and only if it is $E_{n}$-singular.

grandes-ecoles 2017 QII.A.2 Linear Transformation and Endomorphism Properties View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. Show that $A$ is $H$-singular if and only if there exist a non-zero vector $X$ in $H$ and a real number $\lambda$ such that $AX = \lambda N$.

grandes-ecoles 2017 QII.A.3 Determinant and Rank Computation View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. Deduce that $A$ is $H$-singular if and only if the matrix $A_{N} = \left(\begin{array}{cc} A & N \\ N^{\top} & 0 \end{array}\right) \in \mathcal{M}_{n+1}(\mathbb{R})$ is singular.

grandes-ecoles 2017 QII.A.4 Matrix Algebra and Product Properties View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,1}(\mathbb{R}), B_{3} \in \mathcal{M}_{1,n}(\mathbb{R})$, $B_{4} \in \mathcal{M}_{1}(\mathbb{R})$ such that: $A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$.

grandes-ecoles 2017 QII.A.5 Determinant and Rank Computation View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Deduce that $\operatorname{det}\left(A_{N}\right) = -N^{\top} A^{-1} N \operatorname{det}(A)$.

grandes-ecoles 2017 QII.A.6 Determinant and Rank Computation View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that if $\operatorname{det}\left(\left(A^{-1}\right)_{s}\right) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.

grandes-ecoles 2017 QII.A.7 Determinant and Rank Computation View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Deduce that if $\operatorname{det}\left(A_{s}\right) = 0$, then there exists a hyperplane $H$ of $E_{n}$ such that $A$ is $H$-singular.

grandes-ecoles 2017 QII.A.8 Projection and Orthogonality View

We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that $A$ is $H$-regular for every hyperplane $H$ of $E_{n}$.

grandes-ecoles 2017 QII.B.1 Determinant and Rank Computation View

We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Show that $A(\mu)$ is invertible for every real $\mu$.

grandes-ecoles 2017 QII.B.2 Determinant and Rank Computation View

We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Calculate $A(\mu)_{s}$ and show that $A(\mu)_{s}$ is singular for $\mu = 1, 1-\sqrt{3}, 1+\sqrt{3}$.

grandes-ecoles 2017 QII.B.3 Projection and Orthogonality View

We consider the matrix $$A = A(\mu) = \left(\begin{array}{ccc} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{array}\right)$$ Determine a hyperplane $H$ such that $A(1)$ is $H$-singular.

grandes-ecoles 2017 QII.C.1 Linear Transformation and Endomorphism Properties View

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$.
Show that $A$ is $F$-singular if and only if there exist a non-zero element $X$ of $F$ and two real numbers $\lambda_{1}$, $\lambda_{2}$ such that $AX = \lambda_{1} N_{1} + \lambda_{2} N_{2}$.

grandes-ecoles 2017 QII.C.2 Determinant and Rank Computation View

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$.
Deduce that $A$ is $F$-singular if and only if the matrix $$A_{N} = \left(\begin{array}{ccc} A & N_{1} & N_{2} \\ N_{1}^{\top} & 0 & 0 \\ N_{2}^{\top} & 0 & 0 \end{array}\right) = \left(\begin{array}{cc} A & N \\ N^{\top} & 0_{2} \end{array}\right) \in \mathcal{M}_{n+2}(\mathbb{R})$$ is singular.

grandes-ecoles 2017 QII.C.3 Matrix Algebra and Product Properties View

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,2}(\mathbb{R}), B_{3} \in \mathcal{M}_{2,n}(\mathbb{R})$ and $B_{4} \in \mathcal{M}_{2}(\mathbb{R})$ such that $$A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$$

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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