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.