Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 2

The characteristic polynomial is $\det (\lambda I-M)=\lambda ({\lambda }^2-2)$. Solving the three eigenvector equations givesEach eigenspace consists of the scalar multiples of its displayed nonzero vector. Their pairwise dot products vanish: $v_{+}\cdot v_{-}=1-2+1=0$ and $v_{\pm }\cdot v_0=1-1=0$. This also agrees with the orthogonal diagonalization of a real symmetric matrix.

The Euclidean norms are $\Vert v_{\pm }\Vert =2$ and $\Vert v_0\Vert =\sqrt{2}$. With normalized eigenvectors as rows,The rows form an orthonormal basis, so $R{R}^T=I$ and $R$ is an orthogonal matrix. The zero eigenvalue means $M{v}_0=0$ for a nonzero vector; thus $M$ is singular and has no matrix inverse.

The upper-left block is a planar rotation matrix. Its real characteristic polynomial is $(\lambda -\cos \theta {)}^2+{\sin }^2\theta$, which is strictly positive for real $\lambda$ because the angle excludes integer multiples of $\pi$. Hence any real eigenvector has $x=y=0$. The remaining equation is $5z=\lambda z$, givingThere are infinitely many nonzero solutions, spanning one independent eigenvector direction.