  In singular value decomposition (SVD) of a complex matrix A, the singular
vectors or the eigenvectors of AA{\dag} and A{\dag}A are unique up to complex
phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal
matrices of phase factors, the phase-factor matrices. Also, the product of
these two phase-factor matrices, or the product of phase factors of the
corresponding singular vectors with the same singular value, is unique. In the
Schmidt decomposition, a phase-factor matrix is a phase rotation operator
acting on a subsystem alone. We summarize here three simple steps to
consistently carry out the SVD and the Schmidt decomposition including the
phase factors.

Numerical Analysis;Numerical Analysis;Quantum Physics