Qrot {robustX}

Rotation Matrix to Specific Direction

Description

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).

Usage

Qrot(p, transpose = FALSE, unit.image = rep(1, p))

Arguments

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” \propto(1,1,1,...,1).

Details

The qr decomposition is used for a Gram-Schmitt basis orthogonalization.

Value

p \times p orthogonal matrix which rotates (1,0,...,0) onto a vector proportional to unit.image.

Author(s)

Martin Maechler

See Also

qr, matrix (and vector) multiplication, %*%.

Examples

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
}