If T is hermitian or skew-hermitian, then two distinct eigenvalues impliy that two eigenvectors are orthogonal.
But, there is another theorem that if T is hermitian or skew hermitian , then there exists eigenvectors foming a orthonormal basis.
Second theorem does not specify about eigenvalues. Am I under assumption that eigenvalues are distinct for second theorem?
If not, what is so special about first theorem? Orthonormality implies orthogonal. If eigenvalues could be same for second theorem, then what does first theorem tell?