  The different orthogonal relationships that exists in the Lowdin
orthogonalizations is presented. Other orthogonalization techniques such as
polar decomposition (PD), principal component analysis (PCA) and reduced
singular value decomposition (SVD) can be derived from Lowdin methods. It is
analytically shown that the polar decomposition is presented in the symmetric
orthogonalization; principal component analysis and singular value
decomposition are in the canonical orthogonalization. The canonical
orthogonalization can be brought in into the form of reduced SVD or vice-versa.
The analytic relation between symmetric and canonical orthogonalization methods
is established. The inter-relationship between symmetric orthogonalization and
singular value decomposition is presented.
