Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 3 / 7D / iv / Solution

TakeThe real eigenvalues are one and two, with eigenlines spanned by $(1,0{)}^T$ and $(1,1{)}^T$ respectively. These independent vectors form a real eigenbasis, so $E$ is a diagonalizable matrix over $\mathbb{R}$. Their inner product is one, and any two nonzero vectors on those eigenlines have nonzero inner product. Rescaling can normalize their lengths but cannot make them orthogonal. Hence no orthonormal eigenbasis exists.