[R] QR decomposition question

[Spencer Graves]

      It does indeed mean a linear dependency between the response 
variables.  For a discussion of an example with 2 linear dependencies, 
see: 

Box, G. E. P., Hunter, W. G., MacGregor, J. F. and Erjavec, J. (1973)
Some problems associated with the analysis of multiresponse data
Technometrics, 15, 33-51
Keywords: Estimation; Linear dependencies; Eigenvalue-Eigenvector 
analysis; Nonlinear models; Chemical kinetics

      There are doubtless more recent references, but I know this paper 
and I can't think of any more recent off the top of my head. 

      hope this helps. spencer graves

Mäkinen Jussi wrote:

>There is N-1 linearly independent columns in the NxN matrix.
>
>Jussi
>
>Jussi Mäkinen
>Analyst, State Treasury, Finance, Finland
>
>
>
>-----Original Message-----
>From: r-help-bounces at stat.math.ethz.ch [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Alex Nu
>Sent: 1. heinäkuuta 2004 16:46
>To: r-help at stat.math.ethz.ch
>Subject: [R] QR decomposition question
>
>
>
> Hi all,
>
> I wonder if this kind of questions are ok in this
>list...
>
> Quick question:
>
> What does it mean than the rank of the QR
> decomposition of a NxN matrix  is  N-1 ?
>
>  m: NxN matrix
>  qr(m)$rank equal to (N-1) 
>
>
> Long version:
> I'm doing a manova on a matrix of 10 variables
> and 16 observations.
>
>  
>
>>dim(tmp)
>>    
>>
>[1] 16 10
>  
>
>>fit <- manova( tmp ~ treatment*mouse )
>>    
>>
>
>  
>
>>results <- summary(fit,test="Wilks")
>>    
>>
>...
>residuals have rank 9 < 10
>
>So the QR decomposition returns with a rank
> of 9. What does this mean ?
>
> Does this necessarily mean a linear dependency
> among the columns of the matrix ?
>
> Thanks
>
> Alex
>
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