TITLE: When is inverting a matrix numerically unstable?
QUESTION [10 upvotes]: What does numerically unstable mean when inverting a matrix and what are the mathematical conditions that cause this problem to arise?

REPLY [5 votes]: Numerical stability for linear algebra operations is usually associated with the matrix's condition number. A way of estimating the condition number is the ratio of the largest eigenvalue to the smallest eigenvalue, or the largest singular value to the smallest singular value.
What this tells you is the relative scale of the matrix. When doing Gaussian elimination, you will be dividing by some of these numbers. So if you ever divide a large number by a really small number, not only does the result get large, but you will have amplified the error as well. Thus the condition number is related to error estimates, with the larger condition number being less numerically "good".