There is a software called Brandmap$^1$ which can return a biplot from a matrix. I am trying to run the same result in R but the coordinates are not the same.

First I input a simple matrix into the software. 

      x1 x2 x3
    a  6  3  7
    b  8  6  7
    c  9  4  2

There were several options of data centering and I chose to center the data by subtracting every number in the matrix from its (row mean*column mean/grand mean).

             x1    x2         x3
    a -1.076923 -1.00  2.0769231
    b -1.288462  0.75  0.5384615
    c  2.365385  0.25 -2.6153846

Then I chose column factorization to create a biplot. I guess it means a covariance biplot which the singular values are totally assigned to the right singular vectors. 

It showed the coordinates:

    	 dim1	 dim2
    x1	-2.81	-0.73
    x2	-0.55	 1.15
    x3	 3.36	-0.42
    		
    a	 1.58	-1.78
    b	 0.75	 2.26
    c	-2.33	-0.48

I tried to calculate the same results in R.

    > P = matrix(c(6,8,9,3,6,4,7,7,2),nrow=3)
    > row.names(P)=c("a","b","c")
    > colnames(P)=c("x1","x2","x3")
    > P
      x1 x2 x3
    a  6  3  7
    b  8  6  7
    c  9  4  2
    > r1 = matrix(rep(1,3))     #row sum
    > c1 = matrix(rep(1,3))     #column sum
    > r = P%*%r1
    > c = t(P)%*%c1
    > L = P - r%*%t(c)/sum(P)   #subtract row mean*column mean/grand mean
    > L
             x1    x2         x3
    a -1.076923 -1.00  2.0769231
    b -1.288462  0.75  0.5384615
    c  2.365385  0.25 -2.6153846
    > S = svd(L)
    > S$v%*%diag(S$d)
               [,1]       [,2]         [,3]
    [1,]  2.8077724  0.7289408 -8.10596e-17
    [2,]  0.5487104 -1.1506159 -8.10596e-17
    [3,] -3.3564829  0.4216750 -8.10596e-17
    > S$u
               [,1]       [,2]      [,3]
    [1,] -0.5420705  0.6105950 0.5773503
    [2,] -0.2577555 -0.7747443 0.5773503
    [3,]  0.7998260  0.1641494 0.5773503

I found that the values in the right vector are the same but with negative sign and all the values in the left vector are multiplied by -2.918.
I am not sure if there is any weighting in the calculation of that software. What kind of adjustment I can try so that I can run the same results in R?

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$^1$ Note from @ttnphns: I suppose this software does correspondense analysis, not just arbitrary biplot. CA is very often used in brand research.