# Least Squares Normal Equations in Explicit Form

I am struggling with the following least squares problem:

Find the minimiser x* $$\in \mathbb{R}^{m}$$ of

$$F(\textbf{x})=||\textbf{b}-A\textbf{x}||_2$$

where $$A \in \mathbb{R}^{(m+1) \times m}$$, $$\textbf{b} \in \mathbb{R}^{m+1}$$ have the structures

$$A = \begin{bmatrix} m\textbf{u}^T \\ I_m \end{bmatrix} , \textbf{b} = \begin{bmatrix} 1 \\ \textbf{u} \end{bmatrix} ,$$

where $$I_m= \begin{bmatrix} 1 && && && \\ && 1 && && \\ && && \ddots && \\ && && && 1 \end{bmatrix} \in \mathbb{R}^{m \times m}, \textbf{u} = \textbf{1}_m = \begin{bmatrix} 1 \\ \dots \\ 1 \end{bmatrix} \in \mathbb{R}^m$$

1. Show that the normal equations have the explicit form $$$$(I + \alpha \textbf{u}\textbf{u}^T)\textbf{x}^{*} = \beta\textbf{u}$$$$ for some constants $$\alpha,\beta$$ which should be identified.
2. Write the inverse of the coefficient matrix in part 1. in the form $$$$(I + \alpha \textbf{u}\textbf{u}^T)^{-1} = I - \gamma\textbf{u}\textbf{u}^{T}$$$$ for some constant $$\gamma$$ which should be identified. Hence find the solution of $$\textbf{x}^*$$ of the least squares problem.
3. Explain why it is not advisable in practice to construct the coefficient matrix in part 1.

Any help on this matter would be greatly appreciated