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