Does there exist a real normal matrix that's not symmetric, antisymmetric, orthogonal, and has distinct singular eigenvalues?

Question

This question points out a matrix on Wikipedia that is real, normal, and neither symmetric, antisymmetric, or orthogonal. However, its singular values are 2, 1, 1. Is there one that has distinct singular values?

Answer 1

No, there is not any such matrix.

• If a real normal matrix has a real spectrum, it is orthogonally diagonalisable and hence symmetric.
• If a real normal matrix has a non-real eigenvalue $\lambda$, then $\bar\lambda$ is also an eigenvalue of this matrix, because non-real eigenvalues of a real matrix must occur in conjugate pairs. But then $|\lambda|$ will be a repeated singular value, because the singular values of a normal matrix are the moduli of the eigenvalues.