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

Proceed by induction on $\dim V$. The result is immediate for dimension zero. In positive dimension, the characteristic polynomial has a root over the complex numbers, so there is a nonzero eigenvector $v$ with eigenvalue $\lambda$. Normalize it to have norm one. Self-adjointness givesso $\lambda$ is real.

The orthogonal complement $v^{\perp }$ is an invariant subspace: if $w\perp v$, thenThe restriction of $T$ to $v^{\perp }$ is again self-adjoint, since the same inner product identity holds on that subspace. By induction it has an orthonormal basis of eigenvectors. Adjoining $v$ givesThe existence of the first eigenvector is the essential use of the complex numbers in this proof of the finite-dimensional spectral theorem.