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) Verify that $S$ is a symmetric matrix that admits only non-negative eigenvalues and then show that $\operatorname { Im } ( A ) = \operatorname { Im } ( S )$. (b) Let $u \in \mathbb { R } ^ { n }$ be an eigenvector of $S$ for an eigenvalue $\lambda > 0$ and let $v = A ^ { \mathrm { T } } u / \sqrt { \lambda } \in \mathbb { R } ^ { p }$. Show that $v$ is an eigenvector of $\tilde { S }$ for the eigenvalue $\lambda$ and $\| v \| _ { 2 } = \| u \| _ { 2 }$.
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) Verify that $S$ is a symmetric matrix that admits only non-negative eigenvalues and then show that $\operatorname { Im } ( A ) = \operatorname { Im } ( S )$.
(b) Let $u \in \mathbb { R } ^ { n }$ be an eigenvector of $S$ for an eigenvalue $\lambda > 0$ and let $v = A ^ { \mathrm { T } } u / \sqrt { \lambda } \in \mathbb { R } ^ { p }$. Show that $v$ is an eigenvector of $\tilde { S }$ for the eigenvalue $\lambda$ and $\| v \| _ { 2 } = \| u \| _ { 2 }$.