Centrality of Lowdin orthogonalizations

TL;DR

This paper explores the relationships and differences among various orthogonalization techniques, particularly focusing on Lowdin orthogonalizations, and analytically connects them with polar decomposition, PCA, and SVD.

Contribution

It analytically establishes the inter-relationships between Lowdin orthogonalizations, polar decomposition, PCA, and SVD, providing a unified understanding of these methods.

Findings

Polar decomposition corresponds to symmetric orthogonalization.

PCA and SVD relate to canonical orthogonalization.

Analytic relations unify different orthogonalization methods.

Abstract

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.