3.I.1D

Give an example of a real $3 \times 3$ matrix $A$ with eigenvalues $-1,(1 \pm i) / \sqrt{2}$. Prove or give a counterexample to the following statements:

(i) any such $A$ is diagonalisable over $\mathbb{C}$;

(ii) any such $A$ is orthogonal;

(iii) any such $A$ is diagonalisable over $\mathbb{R}$.

*Typos? Please submit corrections to this page on GitHub.*