  The constrained orthogonal Procrustes problem is the least-squares problem
that calls for a rotation matrix that optimally aligns two corresponding sets
of points in d-dimensional Euclidean space. This problem generalizes to the
so-called Wahba's problem which is the same problem with nonnegative weights.
Given a dxd matrix M, solutions to these problems are intimately related to the
problem of finding a dxd rotation matrix U that maximizes the trace of UM,
i.e., that makes UM a matrix of maximal trace over rotations, and it is well
known this can be achieved with a method based on the computation of the
singular value decomposition (SVD) of M. As the main goal of this paper, we
characterize dxd matrices of maximal trace over rotation matrices in terms of
their eigenvalues, and for d = 2, 3, we show how this characterization can be
used to determine whether a matrix is of maximal trace over rotation matrices.
Finally, although depending only slightly on the characterization, as a
secondary goal of the paper, for d = 2, 3, we identify alternative ways, other
than the SVD, of obtaining solutions to the aforementioned problems.
