atex">\(\hat\mu_k^{\text{new}}\)</span>, note again the training sample classification does not change, we have</p>
<div class="arithmatex">\[\begin{eqnarray}
    \hat\mu_k^{\text{new}} &amp;=&amp; \sum_{g_i=k}\frac{\hat B^Tx_i}{N_k}\non\\
    &amp;=&amp;\hat B^T\sum_{g_i=k}\frac{x_i}{N_k}\non\\
    &amp;=&amp;\hat B^T \hat\mu_k.\non
\end{eqnarray}\]</div>
<p>Third, by definition of <span class="arithmatex">\(\hat\Sigma^{\text{new}}\)</span> and result above, we have</p>
<div class="arithmatex">\[\begin{eqn