Orthogonal Procrustes Problem

For real matrices , the orthogonal Procrustes problem is the least squares problem

(1)

where is the Frobenius norm. On writing

(2)

as a singular value decomposition, expanding the squared Frobenius norm gives

(3)

Maximizing the matrix trace therefore gives an optimizer

(4)

and minimum value

(5)

where are the singular values of . If is a nonsingular matrix, the optimizer is unique; if it is a singular matrix, the optimizer need not be unique (Schönemann 1966).

For the constrained problem in which must be a proper rotation, define

(6)

Then is a solution. Applied to centered point sets, this construction is the Kabsch algorithm.