# A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm

arXiv: https://arxiv.org/abs/1902.03138 · HTML: https://mlsanity.com/paper/1902.03138/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2019.

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## Abstract (arXiv)

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two matrices of the same order. Over past decades, the algorithm of choice for solving this problem has been the Kabsch-Umeyama algorithm, which is effectively no more than the computation of the singular value decomposition of a particular matrix. Its justification, as presented separately by Kabsch and Umeyama, is not totally algebraic since it is based on solving the minimization problem via Lagrange multipliers. In order to provide a more transparent alternative, it is the main purpose of this paper to present a purely algebraic justification of the algorithm through the exclusive use of simple concepts from linear algebra. For the sake of completeness, a proof is also included of the well known and widely used fact that the orientation-preserving rigid motion problem, i.e., the least-squares problem that calls for an orientation-preserving rigid motion that optimally aligns two corresponding sets of points in d-dimensional Euclidean space, reduces to the constrained orthogonal Procrustes problem.