array}
    \hat\Sigma^{\text{new}} &amp;=&amp; \sum_{k=1}^K\sum_{g_i=k}(\hat B^Tx_i-\hat\mu_k^{\text{new}})(\hat B^Tx_i-\hat\mu_k^{\text{new}})^T/(N-K)\non\\
    &amp;=&amp;\frac{1}{N-K}\sum_{k=1}^K\sum_{g_i=k}\hat B^T(x_i-\hat\mu_k)(x_i - \hat \mu_k)^T\hat B\non\\
    &amp;=&amp;\hat B^T\left[\frac{1}{N-K}\sum_{k=1}^K\sum_{g_i=k}(x_i-\hat\mu_k)(x_i - \hat \mu_k)^T\right]\hat B\non\\
    &amp;=&amp;\hat B^T\hat\Sigma\hat B.\non
\end{eqnarray}\]</div>
<p>Therefore, the new linear discriminant function is</p>