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 $n , p$ 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 $\epsilon$.