  In this paper, we apply the hierarchical modeling technique and study some
numerical linear algebra problems arising from the Brownian dynamics
simulations of biomolecular systems where molecules are modeled as ensembles of
rigid bodies. Given a rigid body $p$ consisting of $n$ beads, the $6 \times 3n$
transformation matrix $Z$ that maps the force on each bead to $p$'s
translational and rotational forces (a $6\times 1$ vector), and $V$ the row
space of $Z$, we show how to explicitly construct the $(3n-6) \times 3n$ matrix
$\tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{\perp}$
(orthogonal complement of $V$) using only $\mathcal{O}(n \log n)$ operations
and storage. For applications where only the matrix-vector multiplications
$\tilde{Q}{\bf v}$ and $\tilde{Q}^T {\bf v}$ are needed, we introduce
asymptotically optimal $\mathcal{O}(n)$ hierarchical algorithms without
explicitly forming $\tilde{Q}$. Preliminary numerical results are presented to
demonstrate the performance and accuracy of the numerical algorithms.
