Finding inverse of an ill-conditioned matrix

In your example 6x6 matrix, the first three columns are linearly dependent; the second python error column and third column are both just scalar multiples of the first column:

>> a=<your example>; >> a(:,1:3)./a(:,[1 1 1]) ans =      1.000000000000000e+00     1.135821455052853e-01    -1.319363866841049e-01      1.000000000000000e+00     1.135821455052859e-01    -1.319363866841050e-01      1.000000000000000e+00     1.135821455052855e-01    -1.319363866841045e-01      1.000000000000000e+00     1.135821455052838e-01    -1.319363866841038e-01      1.000000000000000e+00     1.135821455052868e-01    -1.319363866841046e-01      1.000000000000000e+00     1.135821455052856e-01    -1.319363866841046e-01 

The inverse in this case doesn't really make sense; it's python error not just an issue with scaling among the different 3x3 submatrices.

Various possibilities:

• There's a bug in whatever generated the example and the correct matrix is reasonably conditioned (though perhaps badly scaled). Applying a simple diagonal scaling sc = diag(1 ./ sqrt(diag(a))) on both sides will usually fix the scaling if that's the only problem.
• The very slight difference in the last digits of the printing above is actually significant and not just rounding noise from the computation of a. I.e., they're actual signal and are what would really determine the result you want. Then you've just run out of precision in the normal representation. You'll have to do some sort of reformulation to make these visible.
• The column dependence is telling you something about the problem, and it's really a lower dimension than the 6x6. Taking into account the symmetry and looking at a1 = a(3:6, 3:6) (dropping the two dependent columns plus the corresponding rows), the result is badly scaled but sc * a1 * sc (with sc as above) is reasonable with a condition number of about 2e6.