[R] How to solve A'A=S for A

[Ralf Engelhorn]

Dear R-helpers,
thank you very much for your immediate and numerous reactions
(also via E-mail).
I see that the problem I encountered lies in the field of
matrix algebra and not R. Beeing a biologist, I think I should
direct further questions to a local statistician.

To give you some feedback on your engagement, I'd like to report,
what I was struggling with: Following a numerical example
from Burnaby 1966 (Growth-invariant discriminant functions and
generalized distances. Biometrics 22:96-110) one of the first
steps is to solve R'R=W^{-1} for R (where all matrices have p x p
order and W^{-1} is the inverse of the within-group covariance
matrix).

When I tried

W <- matrix(c(1,0,2,0,4,2,2,2,6),3,3)
W.inv <- solve(W)
A.chol <- chol(W.inv);A.chol
         [,1]      [,2]       [,3]
[1,] 2.236068 0.4472136 -0.8944272
[2,] 0.000000 0.5477226 -0.1825742
[3,] 0.000000 0.0000000  0.4082483

I received a matrix A.chol which was different from R given
in Burnaby's example

R <- matrix(c(1,0,-2,0,0.5,-0.5,0,0,1),3,3);R
     [,1] [,2] [,3]
[1,]    1  0.0    0
[2,]    0  0.5    0
[3,]   -2 -0.5    1

The guidance the author provides for finding R is:
"The matrix R may be computed during the process of inverting W (it is not
really necessary to complete the inversion)."
The only way that I know to compute the inverse of a matrix is via the
matrix of cofactors (Searle 1982). But this matrix
did not resemble the R that I was looking for.

The symmetric solution (A=A'=KD^{1/2}K', where K has eigenvectors and D
has eigenvalues) suggested by Jan de Leeuw yields results (discriminant
function coefficients, D^2) as expected, though a matrix calculated in an
intermediate step differs from its counterpart in the example.

To me it looks like having to get deeper into matrix algebra to solve
this.