We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Show that there exist $U \in \mathscr { M } _ { n , k } ( \mathbb { R } )$ and $\Lambda = \operatorname { diag } \left( \lambda _ { 1 } , \ldots , \lambda _ { k } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$ such that $S = U \Lambda U ^ { \mathrm { T } }$ with $\lambda _ { 1 } \geqslant \ldots \geqslant \lambda _ { k } > 0$ and $U ^ { \mathrm { T } } U = I _ { k }$.
(b) Show that $\operatorname { Im } ( S ) = \operatorname { Im } ( U )$ and that $U U ^ { \mathrm { T } }$ is the matrix of the orthogonal projection onto $\operatorname { Im } ( U )$ in $\mathbb { R } ^ { n }$.
(c) By setting $V = A ^ { \mathrm { T } } U D \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ where $D = \operatorname { diag } \left( 1 / \sqrt { \lambda _ { 1 } } , \ldots , 1 / \sqrt { \lambda _ { k } } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$, show that $V ^ { \mathrm { T } } V = I _ { k }$ and $\tilde { S } = V \Lambda V ^ { \mathrm { T } }$.