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 $A \tilde { x } = b$. For all $x \in \mathbb { R } ^ { N }$, express $\| x - \tilde { x } \| _ { A } ^ { 2 } = \langle x - \tilde { x } , A ( x - \tilde { x } ) \rangle$ in terms of $J ( \tilde { x } )$ and $J ( x )$ and deduce that $\tilde { x }$ is the unique minimizer of $J$ on $\mathbb { R } ^ { N }$, that is, $J ( \tilde { x } ) \leq J ( x )$ for all $x \in \mathbb { R } ^ { N }$, and that $\tilde { x }$ is the only point satisfying this property.
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 $A \tilde { x } = b$.
For all $x \in \mathbb { R } ^ { N }$, express $\| x - \tilde { x } \| _ { A } ^ { 2 } = \langle x - \tilde { x } , A ( x - \tilde { x } ) \rangle$ in terms of $J ( \tilde { x } )$ and $J ( x )$ and deduce that $\tilde { x }$ is the unique minimizer of $J$ on $\mathbb { R } ^ { N }$, that is, $J ( \tilde { x } ) \leq J ( x )$ for all $x \in \mathbb { R } ^ { N }$, and that $\tilde { x }$ is the only point satisfying this property.