And the the eigenvectors form an orthonormal basis.

No need for Gram-Schmidt orthonormalisation process here as it is not required for this vectors to be orthonormal.

If an eigenspace has dimension greater than 2, two vectors of not necessarily are orthogonal, so you'll need Gram-Schmidt or another tool.

Only normal matrices are unitarily diagonaliseable - by any unitary matrix whose column vectors are eigenvectors of A of the original (normal) matrix.

And the the eigenvectors form an orthonormal basis.

Fernando Revilla