Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ where $n , p$ and $k$ are strictly positive integers, $k \leqslant \min ( n , p )$. We consider the decomposition $A = U \Sigma V ^ { T }$ constructed in the first part. Let $l \in \mathbb { N } ^ { * }$ and $\widetilde { V } \in \mathscr { M } _ { p , l } ( \mathbb { R } )$ be such that $l < k$ and $\widetilde { V } ^ { \mathrm { T } } \widetilde { V } = I _ { l }$. We denote by $\left( \tilde { v } _ { 1 } , \ldots , \tilde { v } _ { l } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { l }$ the family of columns of $\widetilde { V }$ and by $\left( v _ { 1 } , \ldots , v _ { k } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { k }$ that of columns of $V$.
(a) Verify that $\left\| A - A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \| A \| _ { F } ^ { 2 } - \left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 }$.
(b) Show that $$\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \sum _ { h = 1 } ^ { k } \left( \lambda _ { h } \sum _ { m = 1 } ^ { l } \left\langle v _ { h } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 } \right)$$ where $\langle , \rangle _ { 2 }$ denotes the usual inner product on $\mathbb { R } ^ { p }$.