Matrix approximations and subspace fitting

The problem of fitting a least-squares linear subspace to a higher dimensional set of points is solved by the use of matrix approximation results, both when the subspace is required to pass through a given point, and when it is not so restricted. In the latter case the subspace is shown to pass through the centroid of the point set. It is noted that the principal axes of the set of points enter into the problem in a natural manner.The author is indebted to Prof. Carl Eckart of the Department of Physics, University of Chicago, for much helpful comment and suggestion. To him are due the example of the last section and various other items throughout.