Theorem. If $\{v_1, v_2, \ldots, v_n\}$ is an orthonormal basis of $\mathbb{C}^n$ and $U$ is a unitary matrix, then $\{Uv_1, Uv_2, \ldots, Uv_n\}$ is also an orthonormal basis.
Proof. Since $U$ preserves inner products:
$$\langle Uv_i, Uv_j \rangle = \langle v_i, v_j \rangle = \delta_{ij}$$
So the transformed vectors remain orthonormal. Since $U$ is invertible, they remain a basis.
Application. If $A$ is diagonalized by an orthonormal eigenbasis $\{v_i\}$ with $Av_i = \lambda_i v_i$, and $U$ is unitary, then $U^*AU$ has the same eigenvalues with orthonormal eigenvectors $\{U^*v_i\}$.
This property is essential in the proof of the spectral theorem and Schur decomposition.
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