Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm



James F. Lawrence, Javier Bernal, Christoph J. Witzgall


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 essentially 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 as 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.
Journal of Research (NIST JRES) -


Constrained Orthogornal Procrustes Problem, Orientation-Preserving Rigid Motion, Kabsch-Umeyama Algorithm, Singular Value Decompostion, Rotation Matrix, trace


Lawrence, J. , Bernal, J. and Witzgall, C. (2019), A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm, Journal of Research (NIST JRES), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed April 24, 2024)
Created October 8, 2019, Updated October 12, 2021