Overview
Given a set of {% m %} factors, and factor returns given by the vector {% \vec{f} %}, and given an invertible matrix {% \textbf{C} %}, we can define the following
{% \textbf{B}' = \textbf{B} \textbf{C}^{-1} %}
{% \vec{f}' = \textbf{C} \vec{f} %}
This defines a new set of factor returns and factor loadings such that we have
{% \vec{r} = \vec{\alpha} + \textbf{B}' \vec{f}' + \vec{\epsilon} %}
Factor Covariance
Given the factor covariance matrix {% \Sigma %} of the initial factor definitions, the transformed factors will have the following covariance matrix.
{% \Sigma ' = \textbf{C} \Sigma \textbf{C}' %}