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