The solution of the problen goes back to Schonemann in 1966 in Psycometrics. What is a simple explanation of the problem and of it solution? It is use in some finance papers for example Meucci, Attilio (et.al) (2014).
Here is a link to Peter H. Schonemann's 1966 paper in Psychometrika based on his Ph.D. thesis:
http://web.stanford.edu/class/cs273/refs/procrustes.pdf
A complete view of the orthoganal procrustes problem (also known as OPP) and its solution is given in
T. Viklands, Algorithms for the weighted orthogonal Procrustes problem and other least squares problems, Ph.D. Thesis, Umea University, Sweden, 2006:
http://www8.cs.umu.se/~viklands/PhD.pdf
An overview of Procrustes problems is given in chapter 1 (see, especially, the OPP and WOPP, starting on page 7).
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