Procrustes methods

The basic Procrustes problem is to transform a matrix \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{1}$\end{document} to \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{1}\mathbf{T}$\end{document} in order to match a target matrix \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{2}$\end{document}. Matching necessitates that \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{1}$\end{document} and \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{2}$\end{document} have the same number of rows identified with the same entities but the columns are unrestricted in type and number. Special cases discussed are when T is an orthogonal, projection, or direction‐cosine matrix. Sometimes, both matrices are transformed and size parameters referring to isotropic and various forms of anisotropic scaling may be incorporated. Procrustes methods may be generalized to cover K transformed matrices \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\mathbf{X}_{1}\mathbf{T}_{1}, \ldots, \mathbf{X}_{\mathrm{K}}\mathbf{T}_{\mathrm{K}}$\end{document}, in which case their average (the group average) is important. Applications are in shape analysis, image analysis, psychometrics etc. Copyright © 2010 John Wiley & Sons, Inc.