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.