Matrix eigenvectors in R and MatLab

Regarding the relationship between eigen and svd:

	  For symmetric matrices, the svd is a solution to the Eigenvalue 
problem.  However, if eigenvectors are not normalized to length 1, then 
the two solutions will not look the same.

	  Another current question asked about the differences in eigenanalysis 
between R and Matlab.  In sum, it appears that R sorts the eigenvalues 
in decreasing order of absolute values while Matlab does not, but Matlab 
normalizes the eigenvectors to length 1 while R does not.
The last is not wholly accurate for R 1.6.2 (it only applies to
eigen(symmetric=FALSE)), and the imminent R 1.7.0 will normalize
the eigenvectors (except in back-compatibility mode).
Spencer Graves

David Brahm wrote:
Mikael Niva <mikael.niva at ebc.uu.se> wrote:

Is there anyone who knows why I get different eigenvectors when I run
MatLab and R?

R orders the eigenvalues by absolute value, which seems sensible; the MatLab
eigenvalues you gave do not seem to be in any particular order.

R does not normalize the eigenvectors (as MatLab does), but you can easily do
so yourself:

R> PA9900<-c(11/24 ,10/53 ,0/1 ,0/1 ,29/43 ,1/24 ,27/53 ,0/1 ,0/1 ,13/43
R>   ,14/24 ,178/53 ,146/244 ,17/23 ,15/43 ,2/24 ,4/53 ,0/1 ,2/23 ,2/43 ,4/24
R>   ,58/53 ,26/244 ,0/1 ,5/43)
R> PA9900<-matrix(PA9900,nrow=5,byrow=T)
R> eig <- eigen(PA9900)

R> eig$values   # Note they are in descending order of absolute value:
[1]  1.2352970  0.3901522 -0.2562860  0.2259411  0.1742592

R> sweep(eig$vectors, 2, sqrt(colSums(eig$vectors^2)), "/")
            [,1]         [,2]        [,3]        [,4]        [,5]
[1,] -0.22500913 -0.499825704 -0.43295788 -0.18537961 -0.17952679
[2,] -0.10826756  0.159919608 -0.17713941 -0.05825639 -0.06137926
[3,] -0.94030246 -0.845706299  0.71911349  0.97075584  0.96165016
[4,] -0.03271669 -0.096681499  0.07518268 -0.11595437 -0.17499009
[5,] -0.22893213  0.005790397  0.50832318  0.08017655  0.09279089

This is the same as the MatLab result you gave, except for 2 things:

1) The column order matches the eigenvalue order, so R's columns are in a
   different order than Matlab's.

2) The sign is different for one of the vectors (my column 3, your 2).  The
   sign of an eigenvector is not well defined, even after normalization.

MatLab> wmat =
MatLab>    -0.2250    0.4330   -0.4998   -0.1795   -0.1854
MatLab>    -0.1083    0.1771    0.1599   -0.0614   -0.0583
MatLab>    -0.9403   -0.7191   -0.8457    0.9617    0.9708
MatLab>    -0.0327   -0.0752   -0.0967   -0.1750   -0.1160
MatLab>    -0.2289   -0.5083    0.0058    0.0928    0.0802
MatLab> 
MatLab> dmat =
MatLab>     1.2353         0         0         0         0
MatLab>          0   -0.2563         0         0         0
MatLab>          0         0    0.3902         0         0
MatLab>          0         0         0    0.1743         0
MatLab>          0         0         0         0    0.2259

   Side note: there is some relation between eigenvectors and svd (singular
value decomposition) which I have not fully grokked yet; if anyone has a simple
explanation I'd be grateful.

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Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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Matrix eigenvectors in R and MatLab

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