Extracting a basis with fixed block inside a matrix

Given U   an n×mn×m matrix of rank n   whose columns are denoted by (uj)j⩽m(uj)j⩽m, several authors have already considered the problem of finding a subset σ⊂{1,…,m}σ⊂{1,…,m} such that (ui)i∈σ(ui)i∈σ span RnRn and Tr((∑i∈σuiuit)−1) is minimized. In this paper, we generalize this problem by selecting arbitrary rank matrices instead of rank 1 matrices. Another generalization is considering the same problem while allowing a part of the matrix to be fixed. The methods of selection employed develop into algorithms.