Assume I have a problem where the solution is a singular matrix. I am given several matrices that are all pretty close to each other (i.e. the element-by-element differences are small) and I'm trying to choose the one that is closest to singularity. In reality, my perfect solution would have been singular and I am trying to find the one closest to it.
What is the best way to measure this? I know comparing the determinants is not a good idea. I thought the condition number would be a good measure, but when I tried it out on an example set of matrices (where the determinant of one was an absolute zero) and the one with the largest condition number did not end end up with the largest condition number.
I'm suspecting the reason the condition number is not giving me a good measure is round off errors in the singular values. So is there any better way to figure this out?
EDIT - In case anyone is interested in the problem I'm working on. I'm using Cayley-Menger determinants. The determinant being zero would indicate that the matrix is a distance matrix.