Distinct eigenvalues of a normal matrix

Given a matrix and a diagonal matrix with known positive diagonal entries, what conditions may I impose on the columns of such that has distinct eigenvalues?

Here are some things that I already know:

If and , then

.

Also, this seems to be the most promising lead I have:

,

where and is an eigenvalue/eigenvector of .

Because is Hermitian, the eigenvectors define an orthonormal basis. Thus the change from the canonical basis to a basis of the eigenvectors is solely a rotation which does not change the geometry of the vectors . Therefore, it seems that any geometrical properties that I might find before the change of basis might still hold, but I don't know how I can prove this.

Thanks