BTW,
The same solution can be found using SVD (Singular Value Decomposition)

example,

## Define the matrix that we want to decompose into the product of a 
matrix and its transform
M<-matrix(c(0.6098601,  0.2557882,   0.1857773,
             0.2557882,  0.5127065,  -0.1384238,
             0.1857773, -0.1384238,   0.9351089 ),
       nrow=3, ncol=3, byrow=TRUE)

## Compute the singular-value decomposition, and construct F from its pieces
SVD=svd(M, nu=3, nv=3)
U=SVD$u
D=diag(SVD$d)
V=SVD$v
U %*% D %*% t(V)
F = U %*% sqrt(diag(SVD$d))

## Test to see of the product of F with its transpose is equal to M
F %*% t(F)  #
           [,1]       [,2]       [,3]
[1,] 0.6098601  0.2557882  0.1857773
[2,] 0.2557882  0.5127065 -0.1384238
[3,] 0.1857773 -0.1384238  0.9351089


/Shawn


p.s.
HOWEVER I would still like to find a solution that gives me a diagonal 
matrix for F.
For example, I would like this result:,

     > F
           [,1]   [,2]  [,3]
    [1,] 0.781  0.000 0.000
    [2,] 0.328  0.637 0.000
    [3,] 0.238 -0.341 0.873



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