mixAK (version 2.2)

rRotationMatrix: Random rotation matrix

Description

Generate a random rotation matrix, i.e., a matrix $\boldsymbol{P} = (p_{i,j})_{i=1,\dots,p, j=1,\dots,p},$ which satisfies

a) $\boldsymbol{P}\boldsymbol{P}' = \boldsymbol{I}$,

b) $\boldsymbol{P}'\boldsymbol{P} = \boldsymbol{I}$,

c) $\mbox{det}(\boldsymbol{P}) = 1$.

Usage

rRotationMatrix(n, dim)

Arguments

n

number of matrices to generate.

dim

dimension of a generated matrix/matrices.

Value

• For n=1, a matrix is returned.

  For n>1, a list of matrices is returned.

Details

For dim = 2, $p_{2,1}$ ($\sin(\theta)$) is generated from Unif(0, 1) and the rest computed as follows: $p_{1,1} = p_{2,2} = \sqrt{1 - p_{2,1}^2}$ ($\cos(\theta)$) and $p_{1,2} = -p_{2,1}$ ($-\sin(\theta)$).

For dim $>$ 2, the matrix $\boldsymbol{P}$ is generated in the following steps: 1) Generate a $p\times p$ matrix $\boldsymbol{A}$ with independent Unif(0, 1) elements and check whether $\boldsymbol{A}$ is of full rank $p$.

2) Computes a QR decomposition of $\boldsymbol{A}$, i.e., $\boldsymbol{A} = \boldsymbol{Q}\boldsymbol{R}$ where $\boldsymbol{Q}$ satisfies $\boldsymbol{Q}\boldsymbol{Q}' = \boldsymbol{I}$, $\boldsymbol{Q}'\boldsymbol{Q} = \boldsymbol{I}$, $\mbox{det}(\boldsymbol{Q}) = (-1)^{p+1}$, and columns of $\boldsymbol{Q}$ spans the linear space generated by the columns of $\boldsymbol{A}$.

3) For odd dim, return matrix $\boldsymbol{Q}$. For even dim, return corrected matrix $\boldsymbol{Q}$ to satisfy the determinant condition.

References

Golub, G. H. and Van Loan, C. F. (1996, Sec. 5.1). Matrix Computations. Third Edition. Baltimore: The Johns Hopkins University Press.

Examples

P <- rRotationMatrix(n=1, dim=5)
print(P)
round(P %*% t(P), 10)
round(t(P) %*% P, 10)
det(P)

n <- 10
P <- rRotationMatrix(n=n, dim=5)
for (i in 1:3){
  cat(paste("*** i=", i, "", sep=""))
  print(P[[i]])
  print(round(P[[i]] %*% t(P[[i]]), 10))
  print(round(t(P[[i]]) %*% P[[i]], 10))
  print(det(P[[i]]))
}