We introduce the map $$\begin{aligned} J : \mathbb { R } ^ { N } & \rightarrow \mathbb { R } \\ x & \mapsto \frac { 1 } { 2 } \langle x , A x \rangle - \langle b , x \rangle \end{aligned}$$ where $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, $b \in \mathbb { R } ^ { N }$, and $\tilde { x }$ is the unique vector satisfying $A \tilde { x } = b$. We denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$. Show that $x _ { k }$ identifies with the projection onto $x _ { 0 } + H _ { k }$ for the norm $\| \cdot \| _ { A }$ associated with the matrix $A$, that is, $$\left\| x _ { k } - \tilde { x } \right\| _ { A } = \min _ { x \in x _ { 0 } + H _ { k } } \| x - \tilde { x } \| _ { A }$$
We introduce the map
$$\begin{aligned} J : \mathbb { R } ^ { N } & \rightarrow \mathbb { R } \\ x & \mapsto \frac { 1 } { 2 } \langle x , A x \rangle - \langle b , x \rangle \end{aligned}$$
where $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, $b \in \mathbb { R } ^ { N }$, and $\tilde { x }$ is the unique vector satisfying $A \tilde { x } = b$. We denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$.
Show that $x _ { k }$ identifies with the projection onto $x _ { 0 } + H _ { k }$ for the norm $\| \cdot \| _ { A }$ associated with the matrix $A$, that is,
$$\left\| x _ { k } - \tilde { x } \right\| _ { A } = \min _ { x \in x _ { 0 } + H _ { k } } \| x - \tilde { x } \| _ { A }$$