Symmetric Orthogonalization with the Inverse Square Root of a Gram Matrix

Summary

The document outlines symmetric, also called Löwdin, orthogonalization for a set of linearly independent column vectors. Starting from a matrix of vectors, it seeks a transformed basis whose columns are orthonormal. The general transformation is formed from the inverse square root of the vectors’ Gram matrix, followed by a unitary rotation; choosing the identity rotation gives the symmetric orthogonalization result.

The answer also describes the Schweinler-Wigner matrix and its role in establishing a nearest-set property: among orthonormal bases, the symmetric construction is presented as closest to the original vectors under an L2-type measure. The explanation is algebraic rather than an applied trading example, and it gives no code or numerical demonstration. It is useful background for quantitative work involving correlated vectors or basis transformations, but the document does not explain implementation choices, numerical stability, or how the procedure compares with alternatives such as QR decomposition.

Key ideas

• Orthogonalization transforms linearly independent columns into an orthonormal basis.
• The inverse square root of the Gram matrix provides the symmetric orthogonalization transform.
• Unitary rotations produce the general family of orthonormalized bases.
• The symmetric choice is characterized as closest to the original vectors under an L2-style criterion.
• The explanation gives no implementation or numerical stability guidance.

Tags

# Demonstration of the Schweinler-Wigner Orthogonalization procedure


# Demonstration of the Schweinler-Wigner Orthogonalization procedure












Can anyone give me a practical demonstration of the Schweinler-Wigner Orthogonalization procedure? The steps of performing it or possibly a code snippet.

The Schweinler-Wigner Orthogonalization procedure is a method for orthogonalizing the columns of a matrix in a symmetrical way.

Details can be found here https://shodhganga.inflibnet.ac.in/bitstream/10603/104432/9/09_chapter%202.pdf

## Answer by Attack68 (score 1, accepted)

https://quant.stackexchange.com/a/50050

The paper you presented gave a thorough description. Although it actually didn't refer to Schweinler-Wignler Orthogonalisation, it referred to Lowdin Orthogonalisation, Symmetric Orthogonalisation and Canonical Orthogonalisation and the determination of the Schweinler-Wigner Matrix.

> The paper can be summarised as:

You assume an initial set of linearly independent (columnwise) vectors: $V$.

You can apply a transformation to a new basis: $Z=VA$, which is orthonormal iff $Z^TZ = I$.

Letting $A = (V^TV)^{-1/2}B$, where $B$ is a unitary matrix gives the general solution. Letting $B=I$ is termed Symmetric Orthogonalisation.

The Schweinler-Wigner Matrix is calculated as $(V^TZ)^2$ where the square is element-wise, and it seems to me to be used only to prove the property claimed in the paper that symmertic orthogonalisation returns the set of orthonormal vectors "nearest" to the original set of vectors in a nearest neigbour sense with an $l2$ norm.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.