# optimality of LDA dimensionality reduction

typically when you do dimensionality reduction using LDA, you select $$n_{class}-1$$ vectors with largest eigenvalues as discriminants given the fact that you only need $$n-1$$ dimensions to classify $$n$$ classes.

But LDA discriminant vectors doesn't guarantee orthogonality (since $$Var_{between}/Var_{within}$$ is not symmetric). It means sometimes the second vector may overlap a lot with the first vector, resulting in worse performance than some other vector with smaller eigen value but e.g. orthogonal to the first vector.

In another word, LDA dimensionality reduction is not optimal, with respect to keeping the LDA classification accuracy.

Am I right? or is there something wrong?