Let be a real orthogonal matrix with a real eigenvalue corresponding to some real eigenvector. Show algebraically that and interpret this result geometrically.
Each of the matrices
has an eigenvalue . 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 are diagonalisable (i) over the real numbers; (ii) over the complex numbers.