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 d by d matrix M, solutions to these problems are intricately related to the problem of finding a d by d 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. In this paper, we analyze matrices of maximal trace and explore the possibility of identifying alternative ways, other than the SVD method, of obtaining solutions. As a result, we identify a characterization of these matrices that for d = 2, 3, can be used to determine whether a matrix is of maximal trace. We also identify an alternative way that does not involve the SVD method for solving 2-dimensional problems, and explore the possibility of doing the same in 3 dimensions.