Since we are given that these vectors are an orthonormal basis for and we want to prove it is an orthonormal basis for , it would appear that we just want to prove that . Self adjoint transformations have several nice properities that you would want to prove first, if you haven't already:

All eigenvalues are real.

If u and v are eigenvectors corresponding to distinct eigenvalues, then u and v are orthogonal.

If are eigenvectors and we restrict the transformation to theorthogonal complementof the span of the restriction is still self adjoint.

That way, we can start with one eigenvector, the restict ourselves to the orthogonal complement of its span, find a second eigenvector, etc. That way you should be able to show that there are, in fact, n independent eigenvectors and so