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 $V$. We assume here that $\lambda _ { l } > \lambda _ { l + 1 }$. (a) For all $l + 1 \leqslant i \leqslant k$ and all $1 \leqslant j \leqslant l$, we set $a _ { i } = \sum _ { m = 1 } ^ { l } \left\langle v _ { i } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$ and $b _ { j } = 1 - \sum _ { m = 1 } ^ { l } \left\langle v _ { j } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$. Show that the $\left( a _ { i } \right)$ and $\left( b _ { j } \right)$ are non-negative real numbers and that we have $\sum _ { i = l + 1 } ^ { k } a _ { i } \leqslant \sum _ { j = 1 } ^ { l } b _ { j }$. (b) Show that $\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } \leqslant \sum _ { h = 1 } ^ { l } \lambda _ { h }$ and that we have equality if and only if $\operatorname { Vect } \left( \left\{ v _ { 1 } , \ldots , v _ { l } \right\} \right) = \operatorname { Im } ( \widetilde { V } )$ where $\operatorname { Vect } ( X )$ denotes the vector subspace spanned by $X \subset \mathbb { R } ^ { p }$. (c) Let $M \in \mathscr { M } _ { n , p } ^ { l } ( \mathbb { R } )$. Show that $\| M - A \| _ { F } ^ { 2 } \geqslant \sum _ { h = l + 1 } ^ { k } \lambda _ { h }$ with equality if and only if $M = U _ { * } \Sigma _ { * } V _ { * } ^ { \mathrm { T } }$ where $\Sigma _ { * } = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda _ { l } } \right) , U _ { * }$ (resp. $V _ { * }$ ) is the matrix formed by the first $l$ columns of $U$ (resp. of $V$ ).
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 $V$.
We assume here that $\lambda _ { l } > \lambda _ { l + 1 }$.
(a) For all $l + 1 \leqslant i \leqslant k$ and all $1 \leqslant j \leqslant l$, we set $a _ { i } = \sum _ { m = 1 } ^ { l } \left\langle v _ { i } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$ and $b _ { j } = 1 - \sum _ { m = 1 } ^ { l } \left\langle v _ { j } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$.
Show that the $\left( a _ { i } \right)$ and $\left( b _ { j } \right)$ are non-negative real numbers and that we have $\sum _ { i = l + 1 } ^ { k } a _ { i } \leqslant \sum _ { j = 1 } ^ { l } b _ { j }$.
(b) Show that $\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } \leqslant \sum _ { h = 1 } ^ { l } \lambda _ { h }$ and that we have equality if and only if $\operatorname { Vect } \left( \left\{ v _ { 1 } , \ldots , v _ { l } \right\} \right) = \operatorname { Im } ( \widetilde { V } )$ where $\operatorname { Vect } ( X )$ denotes the vector subspace spanned by $X \subset \mathbb { R } ^ { p }$.
(c) Let $M \in \mathscr { M } _ { n , p } ^ { l } ( \mathbb { R } )$. Show that $\| M - A \| _ { F } ^ { 2 } \geqslant \sum _ { h = l + 1 } ^ { k } \lambda _ { h }$ with equality if and only if $M = U _ { * } \Sigma _ { * } V _ { * } ^ { \mathrm { T } }$ where $\Sigma _ { * } = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda _ { l } } \right) , U _ { * }$ (resp. $V _ { * }$ ) is the matrix formed by the first $l$ columns of $U$ (resp. of $V$ ).