On best rank n matrix approximations

We consider the problem of approximating matrices by matrices of rank n in the |·|p,q norm. Among other results, we prove that if A is a totally positive matrix, and |A|1,1 is the norm given by the sum of the absolute values of the entries of the matrix, then a best rank n approximation to A in this norm is given by a matrix that agrees with A on n rows and n columns.