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A normal matrix over $\mathbb C$ with all eigenvalues real is hermitian (using diagonalization) .But a normal matrix with real eigenvalues is symmetric, is this statement true?

I think, all normal matrices are not diagonalizable i.e rotation matrix .please someone explain over $\mathbb R$,are these statements true?

Thanks.

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A normal matrix over $\mathbb C$ is hermitian AKA self adjoint iff it has real eigenvalues. This is true. However, it does not necessarily imply that the matrix is symmetric.

All normal matrices are diagonalizable with respect to a unitary matrix over $\mathbb C$. However, in the real case, this may not be true because the unitary matrix may involve complex numbers.

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