Past exam of the linear algebra course of the University of Oxford / 2017 / 3 / a / iii / Solution

Let $e_1,\dots ,e_d$ be the given orthonormal basis, with $T{e}_j={\lambda }_j{e}_j$. From the defining identity of the adjoint operator,These coordinates prove $T^{\ast }{e}_j={\overline{\lambda }}_j{e}_j$ without assuming a matrix formula. Thereforefor every basis vector. Thus $T$ is a normal operator, andThe condition of possessing an orthonormal basis of eigenvectors is already part of the hypothesis; real eigenvalues alone would not imply self-adjointness for an arbitrary linear transformation.