Suppose we compute the eigen-decomposition for each \(\hat{\Sigma}_k=\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T\), then ingredients for \(\delta_k(x)\) is:
\((x-\hat{\mu}_k)^T\hat{\Sigma}_k^{-1}(x - \hat{\mu}_k)=\left[\mathbf{U}_k^T (x - \hat{\mu}_k)\right]^T \mathbf{D}_k^{-1} \left[\mathbf{U}_k^T(x-\hat{\mu}_k)\right]\)
\(\log |\hat{\Sigma}_k|=\Sigma_l \log d_{kl}\)
the LDA classifier can be implemented by the following steps:
Sphere the data w.r.t the covariance estimate \(\hat{\Sigma}: X^{*}=\mathbf{D}^{-1/2}\mathbf{U}^TX\)
Classify to the closest class centroid in the transformed space, modulo the effect of the class prior probabilities \(\pi_k\).