Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 9

A real normal matrix satisfies $M{M}^T=M^{T}M$. For a symmetric matrix, $M^T=M$ and both products equal $M^2$. For an antisymmetric matrix, $M^T=-M$ and both equal $-M^2$. For an orthogonal matrix, both products are the identity. Thus each of the three classes consists of normal matrices.

The upper $2\times 2$ block has trace zero and determinant $-{\cos }^{2}2\theta -{\sin }^{2}2\theta =-1$, so its characteristic polynomial is ${\lambda }^2-1$. The third coordinate has eigenvalue two. Angle-addition identities give the normalized eigenvectorsThese formulas apply to every real $\theta$, including values where an expression obtained by dividing by $\sin 2\theta$ would fail. Geometrically the planar block is a reflection: it fixes the axis at angle $\theta$ and reverses its perpendicular direction.

The dot product is ${\mathbf{u}}_{+}\cdot {\mathbf{u}}_{-}=-\cos \theta \sin \theta +\sin \theta \cos \theta =0$. Each planar vector is perpendicular to ${\mathbf{u}}_3$ because its third component vanishes. Each has norm one, so the three vectors form an orthonormal eigenbasis. This also follows from the spectral theorem for the real symmetric coefficient matrix.