On the almost rank deficient case of the least squares problem

This paper studies properties of the solutions to overdetermined systems of linear equations whose matrices are almost rank deficient. Let such a system be approximated by the system of rankr which is closest in the euclidean matrix norm. The residual of the approximate solution depends on the scaling of the independent variable. Sharp bounds are given for the sensitivity of the residual to the scaling of the independent variable. It turns out that these bounds depend critically on a few factors which can be computed in connection with the singular value decomposition. Further the influence from the scaling on the pseudo-inverse solution of a rank deficient system is estimated.