Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 40C / c / Solution

If $A$ is a real symmetric matrix, the finite-dimensional spectral theorem lets us choose the normalized left eigenvector and right eigenvector to be the same vector. Thus

More generally, the unitary diagonalization of a normal matrix shows that a real normal matrix also has coincident normalized left and right eigenspaces for each real eigenvalue. Since the question assumes $\lambda \in \mathbb{R}$, one can again choose $u=v$, and every such simple eigenvalue has