PCA and LDA are two of the most widely used factor analysis tools. PCA construct the orthogonal basis that capture the maximum variance of the input space, $AP=PD$, where each column of $P$ is an eigenvector.
Since each column is orthogonal by definition, we can project the low-dimension data back to the original dimension. This nice property can serve as a "anomaly detection" tool.
However, LDA does not have this property. The basis we build are not orthogonal anymore. Therefore, I think it does not make sense to project the data back from the low dimension manifold.
I did some experiments: if I back-project the low-dimension 'features' back to the original dimension, the reconstructed data will be hugely different from raw data.
So, can I say that LDA is more a dimension-reduction tool? Surely we can do classification in the low-dimension feature space, but that's not what I want to discuss in this post.