Paper 1, Section II, B

Let $R$ be a real orthogonal $3 \times 3$ matrix with a real eigenvalue $\lambda$ corresponding to some real eigenvector. Show algebraically that $\lambda=\pm 1$ and interpret this result geometrically.

Each of the matrices

$M=\left(\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right), \quad N=\left(\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right), \quad P=\frac{1}{3}\left(\begin{array}{rrr} 1 & -2 & -2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{array}\right)$

has an eigenvalue $\lambda=1$. Confirm this by finding as many independent eigenvectors as possible with this eigenvalue, for each matrix in turn.

Show that one of the matrices above represents a rotation, and find the axis and angle of rotation. Which of the other matrices represents a reflection, and why?

State, with brief explanations, whether the matrices $M, N, P$ are diagonalisable (i) over the real numbers; (ii) over the complex numbers.

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