R help - Aug 2003 - A question on orthogonal basis vectors

Hey, R-listers,

I have a question about determining the orthogonal 
basis vectors.
In the d-dimensinonal space, if I already know
the first r orthogonal basis vectors, should I be
able to determine the remaining d-r orthognal basis
vectors automatically?

Or the answer is not unique?

Thanks for your attention.

Fred

The answer is certainly not unique.  Your email doesn't say
whether you are asking about principal components or simply
Gram-Schmidt orthogonalization.

-  tom blackwell  -  u michigan medical school  -  ann arbor  -

On Wed, 13 Aug 2003, Feng Zhang wrote:
> I have a question about determining the orthogonal
> basis vectors.
> In the d-dimensinonal space, if I already know
> the first r orthogonal basis vectors, should I be
> able to determine the remaining d-r orthognal basis
> vectors automatically?
>
> Or the answer is not unique?
>
> Thanks for your attention.
>
> Fred

Dear Fred,

If I understand correctly what you want, the answer is not unique. Think 
about the 3D case where you start with one vector. (I assume, by the way, 
that you mean orthonormal and that you mean unique up to a reflection.) 
There are infinitely many pairs of orthonormal basis vectors for the plane 
orthogonal to the initial vector. On the other hand, picking an arbitrary 
orthonormal basis isn't hard: The Gram-Schmidt method does this, for
example.

I hope that this helps,
  John

At 09:16 AM 8/13/2003 -0500, Feng Zhang wrote:
>Hey, R-listers,
>
>I have a question about determining the orthogonal
>basis vectors.
>In the d-dimensinonal space, if I already know
>the first r orthogonal basis vectors, should I be
>able to determine the remaining d-r orthognal basis
>vectors automatically?
>
>Or the answer is not unique?
>
>Thanks for your attention.
>
>Fred
-----------------------------------------------------
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario, Canada L8S 4M4
email: jfox at mcmaster.ca
phone: 905-525-9140x23604
web: www.socsci.mcmaster.ca/jfox

On 13 Aug 2003 at 10:50, John Fox wrote:
> Dear Fred,
> 
> If I understand correctly what you want, the answer is not unique. Think 
> about the 3D case where you start with one vector. (I assume, by the way, 
> that you mean orthonormal and that you mean unique up to a reflection.) 
> There are infinitely many pairs of orthonormal basis vectors for the plane 
> orthogonal to the initial vector. On the other hand, picking an arbitrary 
> orthonormal basis isn't hard: The Gram-Schmidt method does this, for
example.
To add to this, 
the qr decomposition is really a version of Gram-Schmidt. So if your 
basis vectors are the columns of X, you can do something like
qr.Q(qr(X), complete=TRUE)

Kjetil Halvorsen
> 
> I hope that this helps,
>   John
> 
> At 09:16 AM 8/13/2003 -0500, Feng Zhang wrote:
> >Hey, R-listers,
> >
> >I have a question about determining the orthogonal
> >basis vectors.
> >In the d-dimensinonal space, if I already know
> >the first r orthogonal basis vectors, should I be
> >able to determine the remaining d-r orthognal basis
> >vectors automatically?
> >
> >Or the answer is not unique?
> >
> >Thanks for your attention.
> >
> >Fred
> 
> -----------------------------------------------------
> John Fox
> Department of Sociology
> McMaster University
> Hamilton, Ontario, Canada L8S 4M4
> email: jfox at mcmaster.ca
> phone: 905-525-9140x23604
> web: www.socsci.mcmaster.ca/jfox
> 
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