Vector space/C/Finite-dimensional/Normal endomorphism/Spectral theorem/Fact/Proof

Proof

Suppose first that ${}u_{1},\ldots ,u_{n}$ is an orthonormal basis of ${}V$, where the ${}u_{i}$ are eigenvectors of ${}\varphi$. The describing matrix ${}M$ is a diagonal matrix, its diagonal entries are the eigenvalues. Due to fact, the adjoint endomorphism is described by the conjugated-transposed matrix. Hence, this is also a diagonal matrix; therefore, it commutes with ${}M$, and ${}\varphi$ is normal.

We prove the converse statement by induction over the dimension of ${}V$. So let ${}\varphi$ be normal. The one-dimensional case is clear. Because of des Fundamental theorem of algebra, there exists an eigenvector ${}v\in V$ of ${}\varphi$, and we may assume that it has norm ${}1$. Due to fact  (2), ${}v$ is also an eigenvector of ${}{\hat {\varphi }}$. This implies by fact that ${}W=\mathbb {C} v^{\perp }$ is invariant under ${}\varphi$. Therefore, ${}\varphi$ is the direct sum of the restrictions ${}\varphi$. Hence, the restriction of ${}\varphi$ to ${}W$ is again normal, and the induction hypothesis yields the claim.

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