How to measure how far a matrix is from being singular?

Question

What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number?

Answer 1

Given a matrix norm induced by a vector norm of your choice, the distance of an invertible matrix $A$ to its nearest singular matrix, i.e. $\min\{\|A-B\|:\ B \text{ is singular}\}$, is known to be $\|A^{-1}\|^{-1}=\|A\|/\kappa(A)$.

Note that this is a concept different from (but closely related to) the condition number $\kappa(A)=\|A\|\|A^{-1}\|$. What the condition number measures is not how "singular" a matrix is in terms of its nearness to singular matrices, but how singular it is in terms of its effect on the relative error in the solution $x$ of $Ax=b$ (relative to the relative error in the coefficient vector $b$ ). For most purposes, what people concern is the condition number rather than the distance to the nearest singular matrix.

Answer 2

I am assuming that your matrix is a $n\times n$ matrix. You could take the rank of the matrix. Its possible values are $0,1,\ldots,n$. The matrix is singular if and only if its rank is smaller than $n$. The rank is $0$ if and only if the matrix is the null matrix, which is the most singular of all matrices.