Rotation Matrix to Specific Direction
Construct the p x p rotation matrix that rotates the
unit vector (1,0,....0), i.e., the x_1-axis,
onto (1,1,1,...1)/sqrt(p), or more generally to
u / ||u|| (u :=unit.image).
Qrot(p, transpose = FALSE, unit.image = rep(1, p))
p |
integer; the dimension (of the vectors involved). |
transpose |
logical indicating if the transposed matrix is to returned. |
unit.image |
numeric vector of length p onto which the unit vector should be rotated; defaults to “the diagonal” prop. to(1,1,1,...,1). |
The qr decomposition is used for a Gram-Schmitt basis
orthogonalization.
p x p orthogonal matrix which rotates
(1,0,...,0) onto a vector proportional to unit.image.
Martin Maechler
Q <- Qrot(6)
zapsmall(crossprod(Q)) # 6 x 6 unity <==> Q'Q = I <==> Q orthogonal
if(require("MASS")) {
Qt <- Qrot(6, transpose = TRUE)
stopifnot(all.equal(Qt, t(Q)))
fractions(Qt ^2) # --> 1/6 1/30 etc, in an almost lower-triagonal matrix
}Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.