Qrot | R Documentation |
Construct the p \times 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/{\left\|u\right\|}
(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 |
The qr
decomposition is used for a Gram-Schmitt basis
orthogonalization.
p \times p
orthogonal matrix which rotates
(1,0,...,0)
onto a vector proportional to unit.image
.
Martin Maechler
qr
, matrix (and vector) multiplication,
%*%
.
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
}
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