# Repeated LDA for finding multiple discriminant dimensions

I am not very familiar with LDA but was interested in using it to find multiple, orthogonal discriminant dimensions for data with 2 class labels. I understand that standard that the point of LDA is to find a single axis that discriminates 2 classes so it will necessarily find a single discriminant dimension but I was wondering whether, having found this axis, I would be able to factor it out of the data

i.e. say $$\beta$$ $$\in \mathbb{R}^{m}$$are the LDA coefficients that project the data $$D \in \mathbb{R}^{n x m}$$ where $$n$$ is the number of samples and $$m$$ is the number of features of each sample, onto the discriminant axis. Then calculate

$$D2 = D - \left( \frac{\beta\cdot(\beta^T\cdot D^T)}{(\beta^T\cdot\beta)} \right) ^T$$

and rerun LDA on $$D2$$ again to find a second orthogonal discriminant dimension. I am interested in using this approach mainly for data visualisation but am not sure whether it make sense? If so is there a name for this method?

Finding the discriminant directions is equivalent to finding the eigen values of the matrix $$S_w^{-1} S_b$$, where $$S_w$$ is the within scatter matrix and $$S_b$$ the between scatter matrix. For $$C$$ classes, this matrix has rank $$C-1$$, which means that you can only find $$C-1$$ discriminant directions.

In your case of only two classes, you cannot find more than one direction.