ot\)</span> the corresponding complement. It is easy to verify that </p>
<div class="arithmatex">\[\begin{eqnarray}
    W^{-1} &amp;=&amp; (\hat U_LD^{-1/2})(\hat U_LD^{-1/2})^T + (\hat U_\bot D^{-1/2})(\hat U_\bot D^{-1/2})^T\non\\
    &amp;:=&amp; U_LU_L^T + U_\bot U_\bot ^T.\non
\end{eqnarray}\]</div>
<p>Therefore, we have </p>
<div class="arithmatex">\[\begin{eqnarray}
    \|x-\bar x_k\|_W^2 &amp;=&amp; \|U_L^T(x-\bar x_k)\|^2 + \|U_\bot^T(x-\bar x_k)\|^2\non\\
    &amp;=&amp;\|z-\bar z_k\|^2+ \|U_\bot^