= Solution
Start with the assumed <basis> of <eigenvectors>, grouping them by <eigenvalue>. In each <eigenspace>, apply the <Gram-Schmidt process>, which replaces a linearly independent list by an orthonormal list with the same span. Every new vector stays within that <eigenspace>, since an <eigenspace> is a vector subspace.
By part (iv), different <eigenspaces> are orthogonal. The combined lists therefore form an <orthonormal basis> of the entire space, still consisting of <eigenvectors>. Let $U$ have these vectors as columns. Then $U$ is a <unitary matrix>, so
$$
\boxed{U^{-1}=U^\dagger,\qquad U^\dagger AU=\operatorname{diag}(\lambda_1,\ldots,\lambda_n).}
$$
This gives the required <unitary diagonalization of a normal matrix>, including repeated <eigenvalues>.
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