Characterization and Computation of Matrices of Maximal Trace Over Rotations

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.