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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.

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

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Another interesting concept in this direction is the pseudospectrum.

(sorry for the one-line answer).

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