runif_sm: Sample from the uniform distribution on the Stiefel manifold.
Description
This function draws a sample from the uniform distribution on the Stiefel manifold.
Usage
runif_sm(num, ip, ir)
Arguments
num
number of observations or sample size.
ip
the first dimension \(p\) of the matrix.
ir
the second dimension \(r\) of the matrix.
Value
an array with dimension num, ip and ir containing a sample of draws from the uniform distribution on the Stiefel manifold.
Details
The Stiefel manifold with dimension \(p\) and \(r\) (\(p \geq r\)) is a space whose points are \(r\)-frames in \(R^p\).
A set of \(r\) orthonormal vectors in \(R^p\) is called an \(r\)-frame in \(R^p\).
The Stiefel manifold is a collection of \(p \times r\) full rank matrices \(X\) such that \(X'X = I_r\).