# Re: [R] QR decomposition question

From: Spencer Graves <spencer.graves_at_pdf.com>
Date: Fri 02 Jul 2004 - 02:11:02 EST

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

Mäkinen Jussi wrote:

>There is N-1 linearly independent columns in the NxN matrix.
>
>Jussi
>
>Jussi Mäkinen
>Analyst, State Treasury, Finance, Finland
>
>
>
>-----Original Message-----
>From: r-help-bounces@stat.math.ethz.ch [mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Alex Nu
>Sent: 1. heinäkuuta 2004 16:46
>To: r-help@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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