Computation of 3-d matrices of maximal trace over rotations

The programs here compute 3x3 matrices of maximal trace over rotation matrices from input matrices. A dxd matrix M is of maximal trace over rotation matrices if given any dxd rotation matrix U, the trace of UM does not exceed that of M. Given a dxd matrix M, the problem of finding among all dxd rotation matrices U one such that UM is of maximal trace over rotation matrices is intricately related to the so-called constrained orthogonal Procrustes problem which is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space. It is well known that computing an optimal U can be achieved with a method based on the computation of the singular value decomposition (SVD) of M.

本程序可基于输入矩阵，计算出在所有旋转矩阵下迹值最大的3×3矩阵。若对于任意d×d旋转矩阵U，矩阵UM的迹均不超过原矩阵M的迹，则称d×d矩阵M在旋转矩阵下具有最大迹。给定d×d矩阵M，在全体d×d旋转矩阵U中寻找到使得UM在旋转矩阵下迹值最大的U的问题，与所谓的约束正交普罗克拉斯提斯问题（constrained orthogonal Procrustes problem）存在紧密关联；该问题属于最小二乘问题范畴，其目标是找到一个旋转矩阵，以最优方式对齐d维欧氏空间中的两组对应点集。众所周知，可通过基于矩阵M的奇异值分解（SVD）的方法求解得到最优的U。