Vector space/K/Finite-dimensional/Normal endomorphism/Eigenvalues/Fact

Let $V$ be a finite-dimensional $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Let

$\phi :V⟶V$

denote a normal endomorphism. Then the following statements hold.

1. $\lambda \in 𝕂$ is an eigenvalue of $\phi$ if and only if $\bar{\lambda }$ is an eigenvalue of $\hat{\phi }$.
2. A vector $v\in V$ is an eigenvector of the eigenvalue $\lambda$ if and only if $v$ is an eigenvector of $\hat{\phi }$ of the eigenvalue$\bar{\lambda }$.

Proof, Write another proof