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 $\Sigma = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda } _ { k } \right)$.

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 $\mathbb { R } ^ { p }$.

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 $V$ ).

We consider the family of matrices $B = \left[ b _ { i , j } \right] _ { 1 \leq i , j \leq n } \in \mathcal { M } _ { n } ( \mathbb { R } )$ satisfying the following three properties (called $M$-matrices):
$$\forall i \in \{ 1 , \ldots , n \} , \left\{ \begin{array} { l } b _ { i , i } > 0 \\ b _ { i , j } \leq 0 \text { for all } j \neq i \\ \sum _ { j = 1 } ^ { n } b _ { i , j } > 0 \end{array} \right.$$
Show that if $B$ is an $M$-matrix, then we have
(a) $B$ is invertible
(b) If $F = {}^{ t } \left( f _ { 1 } , \ldots , f _ { n } \right)$ has all positive coordinates, then $B ^ { - 1 } F$ also,
(c) all coefficients of $B ^ { - 1 }$ are positive.

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 $\operatorname { Im } ( W ) ^ { \perp }$.

By applying the previous results to $A _ { n } + \varepsilon I _ { n }$ with $\varepsilon > 0$, show that all coefficients of $A _ { n } ^ { - 1 }$ are positive.

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.

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 $\mathscr { M } _ { p } ( \mathbb { R } )$.

Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that there exists a unique family of real numbers $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ satisfying
$$\left\{ \begin{array} { l } - \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\ u _ { 0 } = u _ { n + 1 } = 0 \end{array} \right.$$

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.

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 $M_n$.

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 $\langle , \rangle _ { F }$.

Show that if $f$ is positive, then $u _ { i } \geq 0$ for all $i \in \{ 0 , \ldots , n + 1 \}$.

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 $\| \tilde { A } - A \| _ { F } \leqslant \epsilon$.

Let $n \in \mathbb { N } ^ { * }$. We define the map $N$ from $\mathcal { M } _ { n } ( \mathbb { R } )$ to $\mathbb { R }$ by the relation:
$$N ( A ) = \sup \left\{ \| A x \| _ { \infty } , \| x \| _ { \infty } \leq 1 \right\}$$
Show that $N$ is a norm on $\mathcal { M } _ { n } ( \mathbb { R } )$ and that if $A = \left[ a _ { i , j } \right] _ { 1 \leq i , j \leq n }$, then
$$N ( A ) = \max _ { i \in \{ 1 , \ldots , n \} } \sum _ { j = 1 } ^ { n } \left| a _ { i , j } \right|$$

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 $\Phi_n^{-1} M_n \Phi_n$.

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 $$\tilde { A } = \tilde { A } V V ^ { \mathrm { T } } P _ { V , W }$$

Let $n \in \mathbb { N } ^ { * }$.
(a) Using the results of questions 14 and 15, show that for the matrix $A _ { n }$ defined at the beginning of part 2, we have:
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$
(b) Deduce that for any diagonal matrix $D _ { n } = \left[ d _ { i , j } \right] _ { 1 \leq i , j \leq n }$ such that $d _ { i , i } \geq 0$ for all $i \in \{ 1 , \ldots , n \}$, we also have
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } + D _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$

Let $A$ be a circulant matrix. Give a polynomial $P \in \mathbb{C}[X]$ such that $A = P(M_n)$.

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 $\epsilon$.

Let $u$ be the unique solution of problem (1) and $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ the family defined by relation (2) for $n \in \mathbb { N } ^ { * }$. Show that there exists a constant $\tilde { C } > 0$, independent of $n$, such that
$$\max _ { 0 \leq i \leq n + 1 } \left| u \left( x _ { i } \right) - u _ { i } \right| \leq \frac { \tilde { C } } { n ^ { 2 } }$$
Hint: one may introduce the vector $X = {}^{ t } \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right)$ where we set $\varepsilon _ { i } = u \left( x _ { i } \right) - u _ { i }$ and compute $A _ { n } X$.

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.

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 $\left\| \rho _ { A } ( \tilde { A } ) \right\| _ { F } \leqslant \sqrt { ( n - k ) k ( p - k ) } \| \tilde { A } - A \| _ { F } \left\| P _ { V , W } - P _ { V , V } \right\| _ { F }$.

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

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 $A$.