deletions | additions       diff --git a/optimalF.tex b/optimalF.tex  index e500632..c6460f4 100644  --- a/optimalF.tex  +++ b/optimalF.tex 
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\end{align*}  Let us denote  \begin{equation}  \begin{split}  X \triangleq U^{1/2}F\left( F^TUF \right)^{-1}F^TU^{1/2},\, \bar B \triangleq BU^{-1/2}.  \end{split}  \label{eq:defX}  \end{equation}  It is easy to verify that $X^2 = X$ and $X=X^T$. Hence $X$ is a symmetric projection matrix. Furthermore, $\rank(X) = \rank(F) = q$.