🚀 go-pugleaf

In article <1993Jan19.113428.9224@ms.uky.edu> clarkp@ms.uky.edu (Clark Wells) writes:
#niall@nmsu.edu (Niall Graham) writes:
#
#
#>The following straight-forward problem has probably been studied
#>extensively.  Any references to it would be greatly appreciated.
#
#>Given a pair of matrices X & Y , find a transformation T for which
#>TX = Y.  In general X & Y are inconsistent, so instead of solving
#>directly for T, we instead seek T which minimizes the sum of the
#>errors squared in the equation TX-Y = 0.
#
#>Thanks in advance,
#>Niall Graham
#>NMSU, Las Cruces
#
#What you have described is known as the Orthogonal Procustes
#Problem, and is an application of the Singular Value
#Decomposition.  A good handling of it can be found in _Matrix
#Computations_ by Gene Golub and Charles van Loan, in their
#Special Topics section.(I have the 2nd ed.)  
#
#I hope the reference is meaningful, and I hope it helps.

Actually the Procrustes problem requires that T be orthogonal.
The present problem can be solved more easily.  The problem
of minimizing the sum of squares of R = TX-Y is equivilent
to minimizing the sum of squares of the components of each
row of R.  Thus if t_i is the i-th row of T and y_i is the
i-th row of Y, the sum of squares of the components of
r_i = t_iX - y_i is to be minimized.  This is an ordinary least
squares problem.

Pete Stewart