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I am preparing for an exam and one of the previous examples deals with LDA: The question is to prove that the total-scatter matrix in Multi-Class LDA can be expressed as the sum of the between-scatter matrix and the within-class scatter matrix.

See the whole question here:

Question

I have found some help online to work through the problem but I am stuck now - I found this slide in a course presentation online (full presentation here www.cse.msu.edu/~cse802/DHSCh3_Component%20AnalysisDiscriminants2012.pptx):

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I am able to follow this until it comes to the split - how is (x-mi+mi-m)^2 the same as (x-mi)^2 + (mi-m)^2?

I feel like I am missing something obvious here ...

I also found this topic here but it didn't seem to go anywhere: Deriving total (within class + between class) scatter matrix

Thanks in advance for any help/advice you can offer.

CrankMuffler
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  • (x-mi+mi-m)^2 = ((x-mi)+(mi-m))^2 = (x-mi)^2 + (mi-m)^2 + 2*(x-mi)*(mi-m) and the linked thread https://stats.stackexchange.com/questions/8625/deriving-total-within-class-between-class-scatter-matrix explains why the last term is zero. Hence, I think it's a duplicate. – amoeba Jan 15 '18 at 12:11
  • @amoeba Thanks - I don't quite follow the other thread - can you detail why this second term is 0? – CrankMuffler Jan 15 '18 at 12:22
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    Because \sum_x(x-mi) is zero. – amoeba Jan 15 '18 at 12:26
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    @amoeba Awesome thanks - I think I get it now! – CrankMuffler Jan 15 '18 at 12:33

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