Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 1D / Solution

Let $c=1/\sqrt{2}$. A concrete real matrix isIts first coordinate has eigenvalue $-1$. The lower block has characteristic polynomial $(\lambda -c{)}^2+c^2$, whose roots are $c\pm ic=(1\pm i)/\sqrt{2}$. Thus this example has all three required eigenvalues. It is also an orthogonal matrix, but orthogonality is not forced by its spectrum, as the second part shows.