Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 331 / 3 / c

For real ${\lambda }_1,{\lambda }_2$, computeThus $L$ is a non-normal matrix. For distinct eigenvalues, takeThese satisfy $L{v}_j={\lambda }_j{v}_j$ but have inner product $v_1^T{v}_2=1$. The eigenvectors are therefore nonorthogonal. When the eigenvalues coincide, only the direction $v_2$ remains as an eigenvector; $L$ is a defective Jordan block.