  This paper studies an optimization problem on the sum of traces of matrix
quadratic forms in $m$ semi-orthogonal matrices, which can be considered as a
generalization of the synchronization of rotations. While the problem is
nonconvex, the paper shows that its semidefinite programming relaxation solves
the original nonconvex problems exactly with high probability, under an
additive noise model with small noise in the order of $O(m^{1/4})$. In
addition, it shows that the solution of a nonconvex algorithm considered in
Won, Zhou, and Lange [SIAM J. Matrix Anal. Appl., 2 (2021), pp. 859-882] is
also its global solution with high probability under similar conditions. These
results can be considered as a generalization of existing results on phase
synchronization.
