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
A question on orthogonal basis vectors
4 messages · Feng Zhang, Thomas W Blackwell, John Fox +1 more
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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