More specificially, I am looking for an answer related to ill conditioning with matrices, and square systems of equations.
In: Mathematics
You can’t take the inverse of a singular matrix. If you multiply something by a singular matrix, you lose information about that thing, so you can’t go backwards by taking the inverse. In a metaphor to regular algebra, it’s like dividing by zero. Any number multiplied by zero collapses to zero, so if you want to undo multiplication by zero, which number do you end up with?
An ill-conditioned matrix is a matrix that is *close* to being singular. It’s technically not singular, but its determinant might be very very small. You can take the inverse of an ill-conditioned matrix and everything works fine, but you need to be careful. That inverse will change *a lot* for small changes in the ill-conditioned matrix. To again use the metaphor to regular algebra, you get something similar when you divide by a number close to 0. For example: 1/0.000001=1,000,000, but 1/0.0000001=10,000,000. The denominators are different by only 0.0000009, but the outcomes are different by 9,000,000. Depending on your application, you may not care, but if the values in your ill-conditioned matrix are measured with any error at all, using its inverse will amplify that error to the point where it will surely drown out any useful information in the result.
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