What I want is the submatrix where all the linearly dependent rows have been eliminated.
I tried implementing this solution, but it doesn't work in the following example:
mm = SparseArray[{{1, 3} ->
0.3828286316736664`, {2, 1} -> -1.`, {2, 3} ->
0.9238193756199684`, {3, 2} -> -0.3828286316736664`, {4, 2} ->
0.9238193756199684`, {5, 3} -> -0.3828286316736664`, {6,
3} -> -0.9238193756199684`, {7, 4} ->
0.0679619061399982`, {7, 5} ->
0.30155693875441497`, {8, 4} -> -0.9976879167925299`, {8, 5} ->
0.9534481699017866`, {9, 4} -> -0.0679619061399982`, {9, 8} ->
0.4709417025109601`, {10, 4} ->
0.9976879167925299`, {10, 8} -> -0.8821643343709143`, {11,
5} -> -0.30155693875441497`, {_, _} -> 0}, {11, 10}]
MatrixRank[mm]
the matrix rank is 6. We now run the recipe in the other answer, but I get the new matrix to have rank 1:
MatrixRank[
mm[[Flatten[
Position[#, Except[0, _?NumericQ], 1, 1] & /@
Last@QRDecomposition@Transpose@mm]]] ]