This is almost right, but a normal matrix is not a matrix that “just scales”, its a normal matrix which can do whatever linear operation it likes.
SVD tells us there exists a factorization A=UΣVT where U and V are orthogonal, and Σ is a “scaling matrix” in the sense that its diagonal. Therefore, using similar logic to you, ATA=VΣUTUΣVT=VΣ2VT which means we rotate, scale by the singular values twice, then rotate back, which is why the eigenvales of this are the squares of the singular values, and the eigenvectors are the right singular vectors.
This is almost right, but a normal matrix is not a matrix that “just scales”, its a normal matrix which can do whatever linear operation it likes.
SVD tells us there exists a factorization A=UΣVT where U and V are orthogonal, and Σ is a “scaling matrix” in the sense that its diagonal. Therefore, using similar logic to you, ATA=VΣUTUΣVT=VΣ2VT which means we rotate, scale by the singular values twice, then rotate back, which is why the eigenvales of this are the squares of the singular values, and the eigenvectors are the right singular vectors.