Hello Haziq,

I hope I've got your problem right: The matrix SB contents the squared Euler distances ( meanA - meanB )² on main diagonal of the matrix. The side elements are acting as pseudo-covariances of the scatters between the different dimensions and are needed  in vTSBv. They define the quality of the separation and they tell you whether a dimension delivers new information or not: if two dimensions of the means are of comparable or equal information, the pseudo covariances are higher and the quality of the seperation decreases.

This distances contents all information about the centres of the classes. After projection you'll loose information: even you won't be able to distinguish between the two classes, because the scatters of the patterns are often not seperable any more in the projected space.

Comparable problems can be found in using ANNs:https://www.researchgate.net/publication/228537242_Artificial_neural_networkshow_to_open_the_black_boxes - chapter 2.3.

regards and success,

Michael

Article Artificial neural networks–how to open the black boxes

You do not want only to separate the mean of different class (in between class mean) but also minimize the within distance. Basically, you want samples with the same label as nearest as possible whereas samples with different label as far as possible. That is why you are computed the best trade-off using the in between scatter matrix and the within scatter matrix.

I can attach a python notebook explaining the difference between PCA and LDA with their implementations to illustrate them. It could be useful probably:

http://nbviewer.ipython.org/github/glemaitre/pattern-recognition-vibot/blob/2014-2015-solution/02-features-extraction/02-features-extraction.ipynb

chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/http://www.doc.ic.ac.uk/~dfg/ProbabilisticInference/IDAPILecture16.pdf

Hi. Look the link.