[R] SVD/Eigenvector confusion

[Philip Warner]

My understanding of SVD is that, for A an mxn matrix, m > n:

     A = UWV*

where W is square root diagonal eigenvalues of A*A extended with zero 
valued rows, and U and V are the left & right eigen vectors of A. But this 
does not seem to be strictly true and seems to require specific 
eigenvectors, and I am not at all sure how these are computed.

Since W should have a zero row at the bottom, which when multiplied by U 
will just remove the last column of U, I have just omitted the last row of 
u from the outset:

eg, in R:

     a <- matrix(c(c(1,2,3),c(5,14,11)),3,2)
     u <- eigen(a %*% t(a))$vectors[,1:2]
     v <- eigen(t(a) %*% a)$vectors
     w <- sqrt(diag(eigen(t(a) %*% a)$values))
     u %*% w %*% t(v)

gives:
            [,1]       [,2]
[1,] -0.9390078  -5.011812
[2,] -3.3713773 -13.734403
[3,] -1.3236615 -11.324660

which seems a little off the mark. The value for v is:

           [,1]       [,2]
[1,] 0.1901389  0.9817572
[2,] 0.9817572 -0.1901389

Where as svd(a)$v is:

            [,1]       [,2]
[1,] -0.1901389  0.9817572
[2,] -0.9817572 -0.1901389

If I substitute this in the above, I get:

     u %*% w %*% t(svd(a)$v)

which returns:

      [,1] [,2]
[1,]    1    5
[2,]    2   14
[3,]    3   11

which is what the SVD should do. I assume there is some rule about setting 
the signs on eigenvectors for SVD, and would appreciate any help.



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