Helmke, UweShayman, Mark A.A Classical problem in matrix analysis and total least squares estimation is that of finding a best approximant of a given matrix by lower rank ones. In this paper the critical points and the local minima are determined for the function on varieties of fixed-rank symmetric, skew-symmetric and rectangular matrices representing the distance to a fixed matrix. Our results extend earlier work of Eckart and Young and Higham.en-USlinear systemsoptimizationCommunicationSignal Processing SystemsTechnical Report