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The constrained orthogonal Procrustes problem is the least-squares problemthat calls for a rotation matrix that optimally aligns two corresponding setsof points in d-dimensional Euclidean space. This problem generalizes to theso-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 theproblem 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 wellknown this can be achieved with a method based on the computation of thesingular value decomposition (SVD) of M. As the main goal of this paper, wecharacterize dxd matrices of maximal trace over rotation matrices in terms oftheir eigenvalues, and for d = 2, 3, we show how this characterization can beused to determine whether a matrix is of maximal trace over rotation matrices.Finally, although depending only slightly on the characterization, as asecondary goal of the paper, for d = 2, 3, we identify alternative ways, otherthan the SVD, of obtaining solutions to the aforementioned problems.

Characterization and Computation of Matrices of Maximal Trace over Rotations