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