grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2018 x-ens-maths1__mp

19 maths questions

Q1 Matrices Matrix Entry and Coefficient Identities View
Give the expression of $\langle A , B \rangle _ { F }$ as a function of the coefficients of $A$ and $B$.
Q2 Matrices Matrix Norm, Convergence, and Inequality View
Let $u \in \mathbb { R } ^ { p }$. Show that $\| A u \| _ { 2 } \leqslant \| A \| _ { F } \| u \| _ { 2 }$.
Q3 Matrices Matrix Norm, Convergence, and Inequality View
Show that $\| A C \| _ { F } \leqslant \| A \| _ { F } \| C \| _ { F }$.
Q4 Matrices Determinant and Rank Computation View
We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Show that $k \leqslant \min ( n , p )$ and that for all $\lambda \in \mathbb { R } ^ { * } , \lambda A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Q5 Matrices Matrix Decomposition and Factorization View
We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Verify that $S$ is a symmetric matrix that admits only non-negative eigenvalues and then show that $\operatorname { Im } ( A ) = \operatorname { Im } ( S )$.
(b) Let $u \in \mathbb { R } ^ { n }$ be an eigenvector of $S$ for an eigenvalue $\lambda > 0$ and let $v = A ^ { \mathrm { T } } u / \sqrt { \lambda } \in \mathbb { R } ^ { p }$. Show that $v$ is an eigenvector of $\tilde { S }$ for the eigenvalue $\lambda$ and $\| v \| _ { 2 } = \| u \| _ { 2 }$.
Q6 Matrices Matrix Decomposition and Factorization View
We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Show that there exist $U \in \mathscr { M } _ { n , k } ( \mathbb { R } )$ and $\Lambda = \operatorname { diag } \left( \lambda _ { 1 } , \ldots , \lambda _ { k } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$ such that $S = U \Lambda U ^ { \mathrm { T } }$ with $\lambda _ { 1 } \geqslant \ldots \geqslant \lambda _ { k } > 0$ and $U ^ { \mathrm { T } } U = I _ { k }$.
(b) Show that $\operatorname { Im } ( S ) = \operatorname { Im } ( U )$ and that $U U ^ { \mathrm { T } }$ is the matrix of the orthogonal projection onto $\operatorname { Im } ( U )$ in $\mathbb { R } ^ { n }$.
(c) By setting $V = A ^ { \mathrm { T } } U D \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ where $D = \operatorname { diag } \left( 1 / \sqrt { \lambda _ { 1 } } , \ldots , 1 / \sqrt { \lambda _ { k } } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$, show that $V ^ { \mathrm { T } } V = I _ { k }$ and $\tilde { S } = V \Lambda V ^ { \mathrm { T } }$.
Q7 Matrices Matrix Decomposition and Factorization View
We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Using the decompositions from question 6, deduce that $$A = U \Sigma V ^ { \mathrm { T } } ,$$ with $\Sigma = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda } _ { k } \right)$.
Q8 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ where $n , p$ and $k$ are strictly positive integers, $k \leqslant \min ( n , p )$. We consider the decomposition $A = U \Sigma V ^ { T }$ constructed in the first part. Let $l \in \mathbb { N } ^ { * }$ and $\widetilde { V } \in \mathscr { M } _ { p , l } ( \mathbb { R } )$ be such that $l < k$ and $\widetilde { V } ^ { \mathrm { T } } \widetilde { V } = I _ { l }$. We denote by $\left( \tilde { v } _ { 1 } , \ldots , \tilde { v } _ { l } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { l }$ the family of columns of $\widetilde { V }$ and by $\left( v _ { 1 } , \ldots , v _ { k } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { k }$ that of columns of $V$.
(a) Verify that $\left\| A - A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \| A \| _ { F } ^ { 2 } - \left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 }$.
(b) Show that $$\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \sum _ { h = 1 } ^ { k } \left( \lambda _ { h } \sum _ { m = 1 } ^ { l } \left\langle v _ { h } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 } \right)$$ where $\langle , \rangle _ { 2 }$ denotes the usual inner product on $\mathbb { R } ^ { p }$.
Q9 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ where $n , p$ and $k$ are strictly positive integers, $k \leqslant \min ( n , p )$. We consider the decomposition $A = U \Sigma V ^ { T }$ constructed in the first part. Let $l \in \mathbb { N } ^ { * }$ and $\widetilde { V } \in \mathscr { M } _ { p , l } ( \mathbb { R } )$ be such that $l < k$ and $\widetilde { V } ^ { \mathrm { T } } \widetilde { V } = I _ { l }$. We denote by $\left( \tilde { v } _ { 1 } , \ldots , \tilde { v } _ { l } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { l }$ the family of columns of $\widetilde { V }$ and by $\left( v _ { 1 } , \ldots , v _ { k } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { k }$ that of columns of $V$.
We assume here that $\lambda _ { l } > \lambda _ { l + 1 }$.
(a) For all $l + 1 \leqslant i \leqslant k$ and all $1 \leqslant j \leqslant l$, we set $a _ { i } = \sum _ { m = 1 } ^ { l } \left\langle v _ { i } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$ and $b _ { j } = 1 - \sum _ { m = 1 } ^ { l } \left\langle v _ { j } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$. Show that the $\left( a _ { i } \right)$ and $\left( b _ { j } \right)$ are non-negative real numbers and that we have $\sum _ { i = l + 1 } ^ { k } a _ { i } \leqslant \sum _ { j = 1 } ^ { l } b _ { j }$.
(b) Show that $\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } \leqslant \sum _ { h = 1 } ^ { l } \lambda _ { h }$ and that we have equality if and only if $\operatorname { Vect } \left( \left\{ v _ { 1 } , \ldots , v _ { l } \right\} \right) = \operatorname { Im } ( \widetilde { V } )$ where $\operatorname { Vect } ( X )$ denotes the vector subspace spanned by $X \subset \mathbb { R } ^ { p }$.
(c) Let $M \in \mathscr { M } _ { n , p } ^ { l } ( \mathbb { R } )$. Show that $\| M - A \| _ { F } ^ { 2 } \geqslant \sum _ { h = l + 1 } ^ { k } \lambda _ { h }$ with equality if and only if $M = U _ { * } \Sigma _ { * } V _ { * } ^ { \mathrm { T } }$ where $\Sigma _ { * } = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda _ { l } } \right) , U _ { * }$ (resp. $V _ { * }$ ) is the matrix formed by the first $l$ columns of $U$ (resp. of $V$ ).
Q10 Matrices Linear System and Inverse Existence View
Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, we denote by $M _ { V , W }$ the matrix of $\mathscr { M } _ { p + k } ( \mathbb { R } )$ defined in blocks by $$M _ { V , W } = \left( \begin{array} { c c } V & I _ { p } \\ O _ { k } & W ^ { \mathrm { T } } \end{array} \right) .$$
We assume here that $W ^ { \mathrm { T } } V$ is an invertible matrix.
(a) Show that $M _ { V , W }$ is invertible. We denote its inverse by $M _ { V , W } ^ { - 1 }$.
(b) Show that the orthogonal complement $\operatorname { Im } ( W ) ^ { \perp }$ of $\operatorname { Im } ( W )$ and $\operatorname { Im } ( V )$ are two supplementary subspaces in $\mathbb { R } ^ { p }$, i.e., $\operatorname { Im } ( W ) ^ { \perp } \oplus \operatorname { Im } ( V ) = \mathbb { R } ^ { p }$. Hint: You may start by verifying that for $z \in \mathbb { R } ^ { p }$, if $z \in \operatorname { Im } ( W ) ^ { \perp }$ then $W ^ { \mathrm { T } } z = 0$.
(c) We define the matrix $$P _ { V , W } = \left( \begin{array} { l l } V & O _ { p } \end{array} \right) M _ { V , W } ^ { - 1 } \binom { I _ { p } } { O _ { k , p } } .$$ Show that $P _ { V , W }$ is the matrix of the projection onto $\operatorname { Im } ( V )$ parallel to $\operatorname { Im } ( W ) ^ { \perp }$.
Q11 Matrices Linear System and Inverse Existence View
Let $q \in \mathbb { N } ^ { * }$. Show that the set of invertible matrices of $\mathscr { M } _ { q } ( \mathbb { R } )$ is an open set and that the map $M \mapsto M ^ { - 1 }$ is continuous on this open set.
Q12 Matrices Projection and Orthogonality View
Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, $P_{V,W}$ denotes the matrix of the projection onto $\operatorname{Im}(V)$ parallel to $\operatorname{Im}(W)^\perp$ (when $W^T V$ is invertible).
Show that there exists a neighborhood $\mathscr { V }$ of $V$ in $\mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $W ^ { \mathrm { T } } V$ is invertible for all $W \in \mathscr { V }$ and the map $W \mapsto P _ { V , W }$ is continuous from $\mathscr { V }$ to $\mathscr { M } _ { p } ( \mathbb { R } )$.
Q13 Matrices Determinant and Rank Computation View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
Let $( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E }$. We consider the curve $\gamma : \mathbb { R } \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } )$ defined by $\gamma ( t ) = ( U + t \bar { U } ) ( \Sigma + t \bar { \Sigma } ) ( V + t \bar { V } ) ^ { \mathrm { T } }$.
(a) Show that the functions $t \mapsto \operatorname { rg } ( U + t \bar { U } ) , t \mapsto \operatorname { rg } ( \Sigma + t \bar { \Sigma } )$ and $t \mapsto \operatorname { rg } ( V + t \bar { V } )$ are constant in a neighborhood of $t = 0$.
(b) Deduce that $\gamma ( t ) \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ in a neighborhood of $t = 0$.
(c) Show that $\gamma$ is infinitely differentiable on $\mathbb { R }$ and give the expression of the derivative $\gamma ^ { \prime } ( 0 )$ of $\gamma$ at 0.
Q14 Matrices Matrix Decomposition and Factorization View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
We denote by $T _ { A } = \left\{ \bar { U } \Sigma V ^ { \mathrm { T } } + U \bar { \Sigma } V ^ { \mathrm { T } } + U \Sigma \bar { V } ^ { \mathrm { T } } \mid ( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E } , \bar { U } ^ { \mathrm { T } } U = \bar { V } ^ { \mathrm { T } } V = O _ { k } \right\}$.
(a) Verify that all elements of $T _ { A }$ are tangent vectors to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ at $A$ and that $T _ { A }$ is a vector subspace of $\mathscr { M } _ { n , p } ( \mathbb { R } )$ whose dimension you will give.
(b) Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$. Show that $N _ { A }$ is the orthogonal subspace to $T _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$ for the inner product $\langle , \rangle _ { F }$.
Q15 Matrices Matrix Decomposition and Factorization View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
Let $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$. We say that $\tilde { A }$ satisfies condition (C) if $$\text { (C) } \operatorname { Im } \left( \tilde { \mathrm { A } } V V ^ { \mathrm { T } } \right) = \operatorname { Im } ( \tilde { \mathrm { A } } ) \text { and } \operatorname { Im } \left( \tilde { \mathrm { A } } ^ { \mathrm { T } } \mathrm { UU } ^ { \mathrm { T } } \right) = \operatorname { Im } \left( \tilde { \mathrm { A } } ^ { \mathrm { T } } \right)$$
(a) Show that if $\tilde { A }$ satisfies condition (C) then $\operatorname { rg } ( \tilde { A } ) \leqslant k$ and $$\operatorname { Im } \left( \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } } \right) ^ { \perp } = \operatorname { ker } ( \tilde { A } )$$
(b) Show that there exists $\epsilon > 0$ such that for all $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$, the matrix $\tilde { A }$ satisfies condition (C) as soon as $\| \tilde { A } - A \| _ { F } \leqslant \epsilon$.
Q16 Matrices Projection and Orthogonality View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$ and $\pi_A$ the orthogonal projection onto $T_A$.
Let $\phi : \mathscr { M } _ { n , p } ( \mathbb { R } ) \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } ) \times \mathscr { M } _ { p , n } ( \mathbb { R } )$ defined by $\phi ( \tilde { A } ) = \left( \tilde { A } V V ^ { \mathrm { T } } , \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } } \right)$ for all $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$.
(a) Identify $\operatorname { ker } ( \phi )$ in terms of $N _ { A }$.
(b) We denote by $\pi _ { A } : \mathscr { M } _ { n , p } ( \mathbb { R } ) \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } )$ the orthogonal projection onto $T _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$. Show that $\phi = \phi \circ \pi _ { A }$.
(c) Let $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ satisfying condition (C). We denote by $W = \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } }$. Show that if $P _ { V , W }$ is the matrix of the projection onto $\operatorname { Im } ( V )$ parallel to $\operatorname { Im } ( W ) ^ { \perp }$ then $$\tilde { A } = \tilde { A } V V ^ { \mathrm { T } } P _ { V , W }$$
Q17 Matrices Projection and Orthogonality View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $\pi_A$ be the orthogonal projection onto $T_A$ in $\mathscr{M}_{n,p}(\mathbb{R})$.
Deduce that there exists $\epsilon > 0$ such that the restriction of $\pi _ { A }$ to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } ) \cap B ( A , \epsilon )$ is injective where $B ( A , \epsilon ) = \left\{ \tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \| \tilde { A } - A \| _ { F } < \epsilon \right\}$ is the open ball of $\mathscr { M } _ { n , p } ( \mathbb { R } )$ centered at $A$ with radius $\epsilon$.
Q18 Matrices Projection and Orthogonality View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$.
Let $\rho _ { A }$ be the orthogonal projection onto $N _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$.
(a) Show that for all $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$, we have $\rho _ { A } ( \tilde { A } ) = \left( I _ { n } - U U ^ { \mathrm { T } } \right) \tilde { A } \left( I _ { p } - V V ^ { \mathrm { T } } \right)$.
(b) Show that $\rho _ { A } ( A B ) = 0$ for all $B \in \mathscr { M } _ { p } ( \mathbb { R } )$.
Let $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ satisfy condition (C).
(c) Show that if $W = \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } }$ $$\rho _ { A } ( \tilde { A } ) = \left( I _ { n } - U U ^ { \mathrm { T } } \right) ( \tilde { A } - A ) V V ^ { \mathrm { T } } \left( P _ { V , W } - P _ { V , V } \right) \left( I _ { p } - V V ^ { \mathrm { T } } \right) .$$
(d) Deduce that $\left\| \rho _ { A } ( \tilde { A } ) \right\| _ { F } \leqslant \sqrt { ( n - k ) k ( p - k ) } \| \tilde { A } - A \| _ { F } \left\| P _ { V , W } - P _ { V , V } \right\| _ { F }$.
Q19 Matrices Matrix Decomposition and Factorization View
Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $T _ { A } = \left\{ \bar { U } \Sigma V ^ { \mathrm { T } } + U \bar { \Sigma } V ^ { \mathrm { T } } + U \Sigma \bar { V } ^ { \mathrm { T } } \mid ( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E } , \bar { U } ^ { \mathrm { T } } U = \bar { V } ^ { \mathrm { T } } V = O _ { k } \right\}$.
Show that $T _ { A }$ is exactly the set of tangent vectors to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ at $A$.