# What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more?

I think that over the years numerical folks (who are faced with this problem more often than the others) must have accumulated a number of interesting ideas. I would like to peek into those ideas, no matter how useless and obscure they turned out to be.

• You should search for: "numerical rank" – Suvrit Mar 19 '15 at 15:30
• Why "heuristics"? The two examples that you mention are rigorously defined measures. – Federico Poloni Mar 19 '15 at 15:31

The distance (in operator norm) from square matrix $A$ to the set of singular matrices is the minimum of the singular values of $A$. This is easy to see from the singular value decomposition.