Given a $d\times d$ matrix $M$, it is well known that finding a $d\times d$ rotation matrix $U$ that maximizes the trace of $UM$, i.e., that makes $UM$ a matrix of maximal trace over rotation matrices, can be achieved with a method based on the computation of the singular value decomposition (SVD) of $M$. We characterize $d\times d$ matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for $d=2,3$, we identify alternative ways, other than the SVD, of computing $U$ so that $UM$ is of maximal trace over rotation matrices.
"Characterization and Computation of Matrices of Maximal Trace Over Rotations." J. Geom. Symmetry Phys. 53 21 - 53, 2019. https://doi.org/10.7546/jgsp-53-2019-21-53