[18871] in s-news-athena
Re: [S] finding vectors orthogonal to a given set? (fwd)
daemon@ATHENA.MIT.EDU (Ranjan Maitra)
Thu Aug 5 11:26:28 1999
Date: Thu, 5 Aug 1999 11:19:45 -0400 (EDT)
From: Ranjan Maitra <maitra@math.umbc.edu>
To: S-news <s-news@wubios.wustl.edu>
Message-Id: <Pine.LNX.4.04.9908051118140.4580-100000@param.math.umbc.edu>
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Many thanks to Douglas Bates for providing me with the following suggestion.
Ranjan
---------- Forwarded message ----------
Date: 05 Aug 1999 10:16:43 -0500
From: Douglas Bates <bates@stat.wisc.edu>
To: Ranjan Maitra <maitra@math.umbc.edu>
Subject: Re: [S] finding vectors orthogonal to a given set?
Ranjan Maitra <maitra@math.umbc.edu> writes:
> I have a set of orthogonal vectors: U_1,U_2,\ldots,U_k in R^n. I want to
> find a set of vectors U_{k+1},U_{k+2},\ldots,U_n such that they together
> form an orthonormal basis in R^n. Can this be done using Splus routines?
The Q matrix from a QR decomposition provides the vectors that you
want. See the documentation for the functions qr and qr.Q.
> v1 <- rnorm(4)
> v2 <- rnorm(4)
> cbind(v1, v2) # bind the columns into a matrix
[,1] [,2]
[1,] -0.010532 1.05105
[2,] -1.036434 -0.55333
[3,] -0.455156 -1.58401
[4,] -0.897544 -0.94624
> qr.Q( qr(cbind(v1, v2)), complete = TRUE ) # orthogonal 4 by 4 matrix
[,1] [,2] [,3] [,4]
[1,] -0.0072901 0.653992 0.74677 -0.12075
[2,] -0.7174213 0.311509 -0.36184 -0.50729
[3,] -0.3150596 -0.689156 0.54161 -0.36395
[4,] -0.6212817 -0.017908 0.13442 0.77176
The last two columns of this result are an orthogonal basis for the
orthogonal complement to the span of v1 and v2.
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