The orthogonal Procrustes problem[1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely maps to .[2][3] Specifically, the orthogonal Procrustes problem is an optimization problem given by
where denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is that Wahba's problem tries to find a proper rotation matrix instead of just an orthogonal one.
The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.
^Gower, J.C; Dijksterhuis, G.B. (2004), Procrustes Problems, Oxford University Press
^Hurley, J.R.; Cattell, R.B. (1962), "Producing direct rotation to test a hypothesized factor structure", Behavioral Science, 7 (2): 258–262, doi:10.1002/bs.3830070216
^Golub, G.H.; Van Loan, C. (2013). Matrix Computations (4 ed.). JHU Press. p. 327. ISBN 978-1421407944.
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