Here is a modified version of qraux.Rd, an edited version of
R-2.14.0/src/library/base/man/qraux.Rd

This gives some details and an example for the case of pivoting.
In this case, it is not true that X = QR; rather X[, pivot] = QR.
It may save some other people bugs and time to have this information.

Tim Hesterberg

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% File src/library/base/man/qraux.Rd
% Part of the R package, http://www.R-project.org<http://www.r-project.org/>
% Copyright 1995-2007 R Core Development Team
% Distributed under GPL 2 or later

\name{QR.Auxiliaries}
\title{Reconstruct the Q, R, or X Matrices from a QR Object}
\usage{
qr.X(qr, complete = FALSE, ncol =)
qr.Q(qr, complete = FALSE, Dvec =)
qr.R(qr, complete = FALSE)
}
\alias{qr.X}
\alias{qr.Q}
\alias{qr.R}
\arguments{
 \item{qr}{object representing a QR decomposition.  This will
   typically have come from a previous call to \code{\link{qr}} or
   \code{\link{lsfit}}.}
 \item{complete}{logical expression of length 1.  Indicates whether an
   arbitrary  orthogonal completion of the \eqn{\bold{Q}} or
   \eqn{\bold{X}} matrices is to be made, or whether the \eqn{\bold{R}}
   matrix is to be completed  by binding zero-value rows beneath the
   square upper triangle.}
 \item{ncol}{integer in the range \code{1:nrow(qr$qr)}.  The number
   of columns to be in the reconstructed \eqn{\bold{X}}.  The default
   when \code{complete} is \code{FALSE} is the first
   \code{min(ncol(X), nrow(X))} columns of the original \eqn{\bold{X}}
   from which the qr object was constructed.  The default when
   \code{complete} is \code{TRUE} is a square matrix with the original
   \eqn{\bold{X}} in the first \code{ncol(X)} columns and an arbitrary
   orthogonal completion (unitary completion in the complex case) in
   the remaining columns.}
 \item{Dvec}{vector (not matrix) of diagonal values.  Each column of
   the returned \eqn{\bold{Q}} will be multiplied by the corresponding
   diagonal value.  Defaults to all \code{1}s.}
}
\description{
 Returns the original matrix from which the object was constructed or
 the components of the decomposition.
}
\value{
 \code{qr.X} returns \eqn{\bold{X}}, the original matrix from
 which the qr object was constructed, provided \code{ncol(X) <= nrow(X)}.
 If \code{complete} is \code{TRUE} or the argument \code{ncol} is greater
than
 \code{ncol(X)}, additional columns from an arbitrary orthogonal
 (unitary) completion of \code{X} are returned.

 \code{qr.Q} returns part or all of \bold{Q}, the order-nrow(X)
 orthogonal (unitary) transformation represented by \code{qr}.  If
 \code{complete} is \code{TRUE}, \bold{Q} has \code{nrow(X)} columns.
 If \code{complete} is \code{FALSE}, \bold{Q} has \code{ncol(X)}
 columns.  When \code{Dvec} is specified, each column of \bold{Q} is
 multiplied by the corresponding value in \code{Dvec}.

 \code{qr.R} returns \bold{R}.  This may be pivoted, e.g. if
 \code{a <- qr(x)} then \code{x[, a$pivot]} = \bold{QR}.
 The number of rows of \bold{R} is
 either \code{nrow(X)} or \code{ncol(X)} (and may depend on whether
 \code{complete} is \code{TRUE} or \code{FALSE}.
}
\seealso{
 \code{\link{qr}},
 \code{\link{qr.qy}}.
}
\examples{
p <- ncol(x <- LifeCycleSavings[,-1]) # not the 'sr'
qrstr <- qr(x)   # dim(x) == c(n,p)
qrstr $ rank # = 4 = p
Q <- qr.Q(qrstr) # dim(Q) == dim(x)
R <- qr.R(qrstr) # dim(R) == ncol(x)
X <- qr.X(qrstr) # X == x
range(X - as.matrix(x))# ~ < 6e-12
## X == Q \%*\% R if there has been no pivoting, as here.
Q \%*\% R

# example of pivoting
x <- cbind(int=1,
          b1=rep(1:0, each=3), b2=rep(0:1, each=3),
          c1=rep(c(1,0,0), 2), c2=rep(c(0,1,0), 2), c3=rep(c(0,0,1),2))
# singular, columns "b2" and "c3" are "extra"
a <- qr(x)
qr.R(a) # columns are int b1 c1 c2 b2 c3
a$pivot
all.equal(x, qr.Q(a) \%*\% qr.R(a))            # no
all.equal(x[, a$pivot], qr.Q(a) \%*\% qr.R(a)) # yes
}
\keyword{algebra}
\keyword{array}

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