# Why are the discriminant axes in linear discriminant analysis (LDA) not orthogonal?

This may be a quite silly question and please correct me if I'm wrong.

The discriminants (discriminant axes) are essentially eigenvectors of $\mathrm{Cov}_\mathrm{within}^{-1} \mathrm{Cov}_\mathrm{between}$, so why are they not orthogonal?

I've found a lot of papers stating they have made improvements to make LDA orthogonal... but I don't get why it's not orthogonal in the first place.

• Because if matrices A and B are both symmetric, the product AB is not necessarily symmetric. And orthogonality of eigenvectors only holds for symmetric matrices. – amoeba Jul 6 '18 at 15:57
• (But reasonable question, +1.) – amoeba Jul 6 '18 at 15:58