Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, we denote by $M _ { V , W }$ the matrix of $\mathscr { M } _ { p + k } ( \mathbb { R } )$ defined in blocks by $$M _ { V , W } = \left( \begin{array} { c c } V & I _ { p } \\ O _ { k } & W ^ { \mathrm { T } } \end{array} \right) .$$ We assume here that $W ^ { \mathrm { T } } V$ is an invertible matrix. (a) Show that $M _ { V , W }$ is invertible. We denote its inverse by $M _ { V , W } ^ { - 1 }$. (b) Show that the orthogonal complement $\operatorname { Im } ( W ) ^ { \perp }$ of $\operatorname { Im } ( W )$ and $\operatorname { Im } ( V )$ are two supplementary subspaces in $\mathbb { R } ^ { p }$, i.e., $\operatorname { Im } ( W ) ^ { \perp } \oplus \operatorname { Im } ( V ) = \mathbb { R } ^ { p }$. Hint: You may start by verifying that for $z \in \mathbb { R } ^ { p }$, if $z \in \operatorname { Im } ( W ) ^ { \perp }$ then $W ^ { \mathrm { T } } z = 0$. (c) We define the matrix $$P _ { V , W } = \left( \begin{array} { l l } V & O _ { p } \end{array} \right) M _ { V , W } ^ { - 1 } \binom { I _ { p } } { O _ { k , p } } .$$ Show that $P _ { V , W }$ is the matrix of the projection onto $\operatorname { Im } ( V )$ parallel to $\operatorname { Im } ( W ) ^ { \perp }$.
Let $p , k$ be two strictly positive integers and $V \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ such that $V ^ { \mathrm { T } } V = I _ { k }$. For all $W \in \mathscr { M } _ { p , k } ( \mathbb { R } )$, we denote by $M _ { V , W }$ the matrix of $\mathscr { M } _ { p + k } ( \mathbb { R } )$ defined in blocks by
$$M _ { V , W } = \left( \begin{array} { c c } V & I _ { p } \\ O _ { k } & W ^ { \mathrm { T } } \end{array} \right) .$$
We assume here that $W ^ { \mathrm { T } } V$ is an invertible matrix.
(a) Show that $M _ { V , W }$ is invertible. We denote its inverse by $M _ { V , W } ^ { - 1 }$.
(b) Show that the orthogonal complement $\operatorname { Im } ( W ) ^ { \perp }$ of $\operatorname { Im } ( W )$ and $\operatorname { Im } ( V )$ are two supplementary subspaces in $\mathbb { R } ^ { p }$, i.e., $\operatorname { Im } ( W ) ^ { \perp } \oplus \operatorname { Im } ( V ) = \mathbb { R } ^ { p }$.\\
Hint: You may start by verifying that for $z \in \mathbb { R } ^ { p }$, if $z \in \operatorname { Im } ( W ) ^ { \perp }$ then $W ^ { \mathrm { T } } z = 0$.
(c) We define the matrix
$$P _ { V , W } = \left( \begin{array} { l l } V & O _ { p } \end{array} \right) M _ { V , W } ^ { - 1 } \binom { I _ { p } } { O _ { k , p } } .$$
Show that $P _ { V , W }$ is the matrix of the projection onto $\operatorname { Im } ( V )$ parallel to $\operatorname { Im } ( W ) ^ { \perp }$.