(a) A Householder transformation (reflection) is given by

$$H=I-\frac{2 u u^{T}}{\|u\|^{2}},$$

where $H \in \mathbb{R}^{m \times m}, u \in \mathbb{R}^{m}$, and $I$ is the $m \times m$ unit matrix and $u$ is a non-zero vector which has norm $\|u\|=\left(\sum_{i=1}^{m} u_{i}^{2}\right)^{1 / 2}$. Show that $H$ is orthogonal.

(b) Suppose that $A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$ with $n<m$. Show that if $x$ minimises $\|A x-b\|^{2}$ then it also minimises $\|Q A x-Q b\|^{2}$, where $Q$ is an arbitrary $m \times m$ orthogonal matrix.

(c) Using Householder reflection, find the $x$ that minimises $\|A x-b\|^{2}$ when

$$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 4 \\ 0 & 2 \\ 0 & 4 \end{array}\right] \quad b=\left[\begin{array}{r} 1 \\ 1 \\ 2 \\ -1 \end{array}\right]$$