Linear Algebra, Python, Matrix Factorization, Numpy.
AᵀA has only real eigenvalues and that eigenvectors corresponding to different eigenvalues are perpendicular. We conclude that we can write AᵀA as VDVᵀ, where V is orthogonal. Since U is also orthogonal, we also have that UᵀU=I. And since AᵀA has nonnegative eigenvalues we can take their root, and