[R] LDA: normalization of eigenvectors (see SPSS)

[Spencer Graves]

	  The following satisfies some of your constraints but I don't know if 
it satisfies all of them.

	  Let V = eigenvectors normalized so t(V) %*% V = I.  Also, let D.5 = 
some square root matrix, so t(D.5) %*% D.5 = Derror, and Dm.5 = 
solve(D.5) = invers of D.5.  The Choleski decomposition ("chol") 
provides one such solution, but you can construct a symmetric square 
root using "eigen".  Then Vstar = Dm.5%*%V will have the property you 
mentioned below.

	  Consider the following:

 > (Derror <- array(c(1,1,1,4), dim=c(2,2)))
      [,1] [,2]
[1,]    1    1
[2,]    1    4
 > D.5 <- chol(Derror)
 > t(D.5) %*% D.5
      [,1] [,2]
[1,]    1    1
[2,]    1    4
 > (Dm.5 <- solve(D.5))
      [,1]       [,2]
[1,]    1 -0.5773503
[2,]    0  0.5773503
 > (t(Dm.5) %*% Derror %*% Dm.5)
      [,1] [,2]
[1,]    1    0
[2,]    0    1

	  Thus,t(Vstar)%*%Derror%*%Vstar =  t(V)%*%t(Dm.5)%*%Derror%*%Dm.5%*%V 
= t(V)%*%V = I.

hope this helps.  spencer graves

Christoph Lehmann wrote:
> Hi dear R-users
> 
> I try to reproduce the steps included in a LDA. Concerning the eigenvectors there is 
> a difference to SPSS. In my textbook (Bortz)
> it says, that the matrix with the eigenvectors 
> 
> V
> 
> usually are not normalized to the length of 1, but in the way that the
> following holds (SPSS does the same thing):
> 
> t(Vstar)%*%Derror%*%Vstar = I
> 
> 
> where Vstar are the normalized eigenvectors. Derror is an "error" or
> "within" squaresum- and crossproduct matrix (squaresum of the p
> variables on the diagonale, and the non-diagonal elements are the sum of
> the crossproducts). For Derror the following holds: Dtotal = Dtreat +
> Derror.
> 
> Since I assume that many of you are familiar with this transformation:
> can anybody of you tell me, how to conduct this transformation in R?
> Would be very nice. Thanks a lot
> 
> Cheers
> 
> Christoph
>