# Non-inductive proof of the spectral theorem (for normal matrices)

Here we present a non-inductive proof of the spectral theorem for normal matrices (which doesn’t use, for instance, proposition 8.6.4 in Artin’s Algebra). (But it does seem to be the same as the proof in Herstein’s Topics in algebra.) It is motivated by a similar direct proof (presented in my class) for Hermitian operators with ${n}$ distinct eigenvalues.

We work in matrix form, so we need to prove that there exists an orthonormal basis of ${\mathbb{C}^n}$ consisting of eigenvectors of a normal matrix ${A}$.