Principal axis transformation/Hermitian/Orthonormal system/Fact

Principal axis theorem

Let $V$ be a finite-dimensional $𝕂$-vector space, endowed with an inner product. Let $\mathrm{\Psi }$ denote an Hermitian form on $V$.

Then there exists an

orthonormal basis of $V$ (with respect to the inner product), which is an orthogonal basis

with respect to $\mathrm{\Psi }$.

Proof, Write another proof