R help - Jun 2003 - understanding eigen(): getting non-normalized eigenvectors

Hi, dear R pros

I try to understand eigen(). I have seen, that eigen() gives the
eigenvectors normalized to unit length.

What shall I do to get the eigenvectors not normalized to unit length?

E.g. take the example:

 A
     
           [,1]       [,2]
  V1  0.7714286 -0.2571429
  V2 -0.4224490  0.1408163

Calculating eigen(A) "by hand" gives the eigenvectors (example from
Backhaus, multivariate analysis):

 0.77143  and 0.25714
-0.42245      0.14082


but even eigen(solve(Derror)%*%Dtreat, symmetric = FALSE, EISPACK =TRUE)
which according to ?eigen should not necessarily give the normalized
eigenvectors give the vectors (such as eigen()):

$vectors
           [,1]      [,2]
[1,]  0.8770963 0.3162278
[2,] -0.4803146 0.9486833


-> how can I replicate the result we get "by hand" (I ask because
for
students it is "nice" to see the same results with R as the results
written in textbooks, derived "manually"?

Thanks a lot
Christoph

-- 
Christoph Lehmann <lehmann at puk.unibe.ch>
University Hospital of Clinical Psychiatry

Eigenvectors are defined only up to a scalar constant (assuming distinct 
eigenvalues).  However, your `by hand' answer does not pass the simple 
test Av = lambda v for some lambda.  So you cannot reproduce incorrect
answers in R!

Your example is unusual: A is of rank 1.

On 9 Jun 2003, Christoph Lehmann wrote:
> Hi, dear R pros
> 
> I try to understand eigen(). I have seen, that eigen() gives the
> eigenvectors normalized to unit length.
> 
> What shall I do to get the eigenvectors not normalized to unit length?
Multiply them by any randomly chosen non-zero scalar!
> E.g. take the example:
> 
>  A
>      
>            [,1]       [,2]
>   V1  0.7714286 -0.2571429
>   V2 -0.4224490  0.1408163
> 
> Calculating eigen(A) "by hand" gives the eigenvectors (example
from
> Backhaus, multivariate analysis):
> 
>  0.77143  and 0.25714
> -0.42245      0.14082
The second is not an eigenvector of A: try it!  They look like rounded
versions of A with a sign error.

-- 
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)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595