A globally convergent flow for computing the best low rank approximation of a matrix

We work in the space of $m$-by-$n$ real matrices with the Frobenius inner product. Consider the following Problem: Given an m-by-n real matrix A and a positive integer k, find the m-by-n matrix with rank k that is closest to A. I discuss a rank-preserving differential equation (d.e.) which solves this problem. If X(t) is a solution of this d.e., then the distance between $X(t)$ and $A$ decreases as t increases; this distance function is a Lyapunov function for the d.e. If $A$ has distinct positive singular values (which is a generic condition) then this d.e. has only one stable equilibrium point. The other equilibrium points are finite in number and unstable. In other words, the basin of attraction of the stable equilibrium point on the manifold of matrices with rank $k$ consists of almost all matrices. This special equilibrium point is the solution of the given problem. Usually constrained optimization problems have many local minimums (most of which are undesirable). So the constrained optimization problem considered here is very special.