According to the answer to this question, the geometrical meanings of normal and diagonalizable matrices are: normal matrices have orthogonal eigenvectors whiles diagonalizable have independent eigenvectors (not necessarily orthogonal).
And according to its definition, a normal matrix shares the same eigenvectors with its adjoint matrix (conjugate transpose).
My question is: how to connect the geometrical meanings of normal matrices to the definition? In other words, how to understand the fact that a matrix shares the same eigenvectors with its adjoint if it has orthogonal eigenvectors, but it may not be so if it only has independent but NOT orthogonal eigenvectors?
On the flip side, how to understand, if a matrix shares the same eigenvectors with its adjoint, then it has orthogonal eigenvectors?