We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and we introduce for all $(\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$ $$J(\tau, R) = \sum_{i=1}^{n} |\boldsymbol{y}_{i} - (R\boldsymbol{x}_{i} + \tau)|^{2} = \|\boldsymbol{y} - g \cdot \boldsymbol{x}\|^{2}$$ where $g = (\tau, R)$. We denote $\overline{\boldsymbol{x}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{x}_{i}$ and $\overline{\boldsymbol{y}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{y}_{i}$.
[(a)] Show that $J(\tau, R) = \left(\sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}} - R(\boldsymbol{x}_{i} - \overline{\boldsymbol{x}})|^{2}\right) + n|\overline{\boldsymbol{y}} - R\overline{\boldsymbol{x}} - \tau|^{2}$.
[(b)] Deduce that for all $R \in \mathrm{SO}_{d}(\mathbb{R})$, the map $\tau \mapsto J(\tau, R)$ from $\mathbb{R}^{d}$ to $\mathbb{R}$ has a unique minimum, denoted $\tau(R)$, which we will express explicitly.
We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and we introduce for all $(\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$
$$J(\tau, R) = \sum_{i=1}^{n} |\boldsymbol{y}_{i} - (R\boldsymbol{x}_{i} + \tau)|^{2} = \|\boldsymbol{y} - g \cdot \boldsymbol{x}\|^{2}$$
where $g = (\tau, R)$. We denote $\overline{\boldsymbol{x}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{x}_{i}$ and $\overline{\boldsymbol{y}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{y}_{i}$.
\begin{itemize}
\item[(a)] Show that $J(\tau, R) = \left(\sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}} - R(\boldsymbol{x}_{i} - \overline{\boldsymbol{x}})|^{2}\right) + n|\overline{\boldsymbol{y}} - R\overline{\boldsymbol{x}} - \tau|^{2}$.
\item[(b)] Deduce that for all $R \in \mathrm{SO}_{d}(\mathbb{R})$, the map $\tau \mapsto J(\tau, R)$ from $\mathbb{R}^{d}$ to $\mathbb{R}$ has a unique minimum, denoted $\tau(R)$, which we will express explicitly.
\end{itemize}