Why SVD matrix is orthogonal matrix?

Question

I am trying to understand Why SVD matrix is orthogonal matrix. Let define SVD as A=UDV', I want to prove that U is a orthogonal matrix. I understand each column of U is the eigen vector of AA'. I also understand that since AA' is symmetric matrix so its eigen vectors are orthogonal. Then we can prove U'U=I. However, to be a orthogonal matrix, U should also have UU'=I. How can we prove UU'=I?