b) Find an orthogonal projection of a vector from $\mathbb{R}^4$ to the basis \{m?,m?\} with
$M = (m_1 m_2) = \begin{pmatrix} 1/\sqrt{2} & 0\\ 0 & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 0 \end{pmatrix}$
by answering the following questions:
i) Compute the 4 \times 4 matrix $MM^T$
[3 marks]
ii) Find a projection of the vector $y = \begin{pmatrix} 2\\3\\1\\2 \end{pmatrix}$ to the space \{m?, m?\} using
the formula
$\text{proj}_{\{m_1, m_2\}} y = MM^T y$
[2 marks]
c) The 3 \times 3 matrix $M$ is given by
$M = \begin{pmatrix} 1 & 0 & 0\\ 1 & 0 & -2\\ 1 & 1 & 2 \end{pmatrix}$
i) If $M$ has one real eigenvalue $\lambda_1 = 1$, show that it has two complex eigen-
values $\lambda_{2,3} = 1 \pm i$. Show all steps clearly.
[5 marks]
ii) Compute the corresponding complex eigenvectors for $\lambda_{2,3} = 1 \pm i$.
[6 marks]
