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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 $$ \begin{equation} (I + \alpha \textbf{u}\textbf{u}^T)\textbf{x}^{*} = \beta\textbf{u} \end{equation} $$ for some constants $\alpha,\beta$ which should be identified.
  2. Write the inverse of the coefficient matrix in part 1. in the form $$ \begin{equation} (I + \alpha \textbf{u}\textbf{u}^T)^{-1} = I - \gamma\textbf{u}\textbf{u}^{T} \end{equation} $$ 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

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