R version 2.9.0 (2009-04-17)
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>  # svd2.r                 September 2007, revised 2008, 2009
>  #
>  # some examples of use of svd to do Singular Value Decomposition
>  #
>  # second example -- another singular design matrix
>  #
>   a <- 3            # number of levels of factor A (alpha's)
>   b <- 4            # number of levels of factor B (beta's)
>   N <- a*b          # number of observations
>   One <- matrix(1,N,1)        # intercept column
>   XA  <- matrix(c(rep(1,b),rep(0,N)),N,a)        # factor A cols
>   XB  <- matrix(rep(diag(1,b),a),N,b,byrow=TRUE) # factor B cols
>   X <- cbind(One,XA,XB)
>  #                    write out design matrix
>   X
      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
 [1,]    1    1    0    0    1    0    0    0
 [2,]    1    1    0    0    0    1    0    0
 [3,]    1    1    0    0    0    0    1    0
 [4,]    1    1    0    0    0    0    0    1
 [5,]    1    0    1    0    1    0    0    0
 [6,]    1    0    1    0    0    1    0    0
 [7,]    1    0    1    0    0    0    1    0
 [8,]    1    0    1    0    0    0    0    1
 [9,]    1    0    0    1    1    0    0    0
[10,]    1    0    0    1    0    1    0    0
[11,]    1    0    0    1    0    0    1    0
[12,]    1    0    0    1    0    0    0    1
>  #                    next svd
>   svd(X)
$d
[1] 4.358899e+00 2.000000e+00 2.000000e+00 1.732051e+00 1.732051e+00
[6] 1.732051e+00 3.594146e-16 1.585726e-16

$u
            [,1]       [,2]        [,3]        [,4]        [,5]        [,6]
 [1,] -0.2886751 -0.1737896  0.36941014 -0.08762684  0.49178314  0.02169989
 [2,] -0.2886751 -0.1737896  0.36941014  0.47927171 -0.07881682 -0.11868672
 [3,] -0.2886751 -0.1737896  0.36941014 -0.09768823 -0.20648316  0.44477153
 [4,] -0.2886751 -0.1737896  0.36941014 -0.29395665 -0.20648316 -0.34778470
 [5,] -0.2886751  0.4068134 -0.03419890 -0.08762684  0.49178314  0.02169989
 [6,] -0.2886751  0.4068134 -0.03419890  0.47927171 -0.07881682 -0.11868672
 [7,] -0.2886751  0.4068134 -0.03419890 -0.09768823 -0.20648316  0.44477153
 [8,] -0.2886751  0.4068134 -0.03419890 -0.29395665 -0.20648316 -0.34778470
 [9,] -0.2886751 -0.2330238 -0.33521125 -0.08762684  0.49178314  0.02169989
[10,] -0.2886751 -0.2330238 -0.33521125  0.47927171 -0.07881682 -0.11868672
[11,] -0.2886751 -0.2330238 -0.33521125 -0.09768823 -0.20648316  0.44477153
[12,] -0.2886751 -0.2330238 -0.33521125 -0.29395665 -0.20648316 -0.34778470
             [,7]        [,8]
 [1,]  0.66711920  0.11303389
 [2,] -0.43595595  0.33881034
 [3,] -0.12384592 -0.30701654
 [4,] -0.10731733 -0.14482769
 [5,] -0.37716245 -0.17533823
 [6,]  0.22827612  0.24068375
 [7,]  0.10057311  0.22373050
 [8,]  0.04831323 -0.28907602
 [9,] -0.28995675  0.06230434
[10,]  0.20767983 -0.57949409
[11,]  0.02327282  0.08328604
[12,]  0.05900410  0.43390372

$v
           [,1]          [,2]          [,3]          [,4]          [,5]
[1,] -0.7947194  8.425252e-19  1.967092e-16  6.287630e-15 -6.237373e-16
[2,] -0.2649065 -3.475791e-01  7.388203e-01 -3.170122e-15  6.045645e-17
[3,] -0.2649065  8.136267e-01 -6.839779e-02 -1.217078e-15  9.326314e-16
[4,] -0.2649065 -4.660476e-01 -6.704225e-01 -3.655073e-15 -1.775916e-16
[5,] -0.1986799 -9.020562e-16  8.326673e-17 -1.517741e-01  8.517934e-01
[6,] -0.1986799 -1.665335e-15 -4.163336e-16  8.301230e-01 -1.365147e-01
[7,] -0.1986799  1.026956e-15 -1.110223e-16 -1.692010e-01 -3.576393e-01
[8,] -0.1986799  1.249001e-15 -1.942890e-16 -5.091478e-01 -3.576393e-01
              [,6]       [,7]       [,8]
[1,] -2.060769e-19 -0.5914052  0.1366053
[2,] -1.735750e-16  0.2528817 -0.4463280
[3,] -1.440273e-17  0.2528817 -0.4463280
[4,] -2.921687e-17  0.2528817 -0.4463280
[5,]  3.758531e-02  0.3385234  0.3097228
[6,] -2.055714e-01  0.3385234  0.3097228
[7,]  7.703669e-01  0.3385234  0.3097228
[8,] -6.023808e-01  0.3385234  0.3097228

>  #                    inner product matrix
>   xpx <- t(X) %*% X
>   print("X'X")
[1] "X'X"
>   xpx
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,]   12    4    4    4    3    3    3    3
[2,]    4    4    0    0    1    1    1    1
[3,]    4    0    4    0    1    1    1    1
[4,]    4    0    0    4    1    1    1    1
[5,]    3    1    1    1    3    0    0    0
[6,]    3    1    1    1    0    3    0    0
[7,]    3    1    1    1    0    0    3    0
[8,]    3    1    1    1    0    0    0    3
>   expx <- eigen(xpx,symmetric=TRUE) # eigenvectors and values of X'X
>   sqrt(expx$values)
[1] 4.358899e+00 2.000000e+00 2.000000e+00 1.732051e+00 1.732051e+00
[6] 1.732051e+00 1.192093e-07 4.214685e-08
>  # done
>   rm(list=ls())
>  q()