Symmetric Factor Orthogonalization Through Eigenvalue Decomposition

# Klein and Chow Orthogonal Transformation - Lowdin Orthogonalization


# Klein and Chow Orthogonal Transformation - Lowdin Orthogonalization












I've doing research on the orthogonal transformation in Orthogonalized Equity Risk Premia and Systematic Risk Decomposition

They borrow a mathematical technique called symmetric orthogonalization from quantum chemistry to to identify the underlying uncorrelated components of the factors and maintains the interpretations of the original factors.

Specifically, given the return $F_{T,K}$, they try to find $F_{T,K}^{\bot}$ by finding $S_{K,K}$. The $S_{K,K}$ which performs symmetric orthogonalization is $M_{K,K}^{-\frac{1}{2}} I_{K,K}$ where $S_{K,K} = O_{K,K}D_{K,K}O_{K,K}^{-1}$, where the $k$-th column of $O_{K,K}$ is the $k$-th eigenvector of the matrix $M_{K,K}$, and $D_{K,K}$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, that is, $D_{K,K} = \lambda_k$, where $k$ goes from 1 to $K$. $M_{K,K}$ is $(T-1)$ times variance-covariance matrix, where

While I have successfully implemented in Python and the test result seems validating my knowledge about the portfolio, there is a few things that I don't fully understand in the methodology.

1) I know that it is not trivial to calculate $M_{K,K}^{-\frac{1}{2}}$ by taking inverse and square root of it, and that's why they perform diagonalization. But why would they diagonalize the matrix $M$ into eigenvalues and eigenvectors (i.e. why do they perform eigendecomposition)? What's so significant about eigenvalues and eigenvectors in this situation?

2) How would you explain the orthogonalization process in layman's terms?

3) It turns out that an orthogonal factor is a linear combination of the original factors, and yet the authors say that the orthogonal factor maintains the interpretability of the original factor. How can a linear combination of factors maintain the interpretation of the original one?

I know that it is a long post with many questions, but these are fundamental ones I am having trouble, and I would greatly appreciate any help.

Thank you very much.

## Answer by Attack68 (score 1, accepted)

https://quant.stackexchange.com/a/41404

I can only say I have had a very brief look into this, without necessarily understanding it, but, is the orthogonalization procedure not related to the Singular Value Decomposition (which answers question 1), which is a main result from linear algebra:

> Any matrix $A \in R^{m \times n}$ can be factored into a singular value decomposition (SVD): $$ A = U S V^T $$ where $U \in R^{m \times m}$ and $V \in R^{n \times n}$ are orthogonal matrices (i.e. $UU^T = VV^T = 1$) and $S \in R^{m \times n}$ is diagonal with with $r=rank(A)$ leading positive diagonal entries. The $p$ diagonal entries of $S$ are usually denoted by $\sigma_i$ for $i=1...p$, where $p = min(m,n)$, and $\sigma_i$ are called the singular values of $A$. The singular values are the square roots of the nonzero eigenvalues of both $AA^T$ and $A^TA$ and they satisfy the property that $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_p$