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 $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Using the decompositions from question 6, deduce that
$$A = U \Sigma V ^ { \mathrm { T } } ,$$
with $\Sigma = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda } _ { k } \right)$.