arle</a> on pages 89–90, where the following facts are obtained, with the aid of formulas derived on pages 29-30:</p>

<ul>
  <li>
    <p>$
\log\left|SVS^T\right| = \log|V| + \log\left|X^TV^{-1}X\right| - \log\left|X^TX\right| + C,\quad
$ where $C$ is a constant that depends on the choice of $S$</p>
  </li>
  <li>
    <p>$S$ can be taken to be $I - XX^{\dagger}$, and in this case $C = 0$.</p>
  </li>
  <li>
    <p>$ (Sy)^T(SVS^T)^{-1}(Sy) = y^TPy \ $
where $P = V^{-1} - V^{-1}X(X^TV^{-1}X)X^TV^{-1}$</p>
 