Least additive modification to a matrix to make it invertible.

Question

Would it be well defined? Say I have a matrix $\bf A$ which I know is not invertible. Is there some way to define the "minimal added contribution" $\bf B$ that will make it invertible?

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Own work: I've been thinking maybe something like this will make it well defined in some sense: $$\min_{\bf B,P}\{\epsilon_1\|(\bf A+ B)P-I\|+ \epsilon_2\|B\|\}$$ Where $\bf P$ is the candidate inverse. But I doubt it would be very well behaved since $\bf B$ and $\bf P$ are multiplied together.

Answer 1

For any matrix $A$, $A + \epsilon I$ will be invertible for sufficiently small nonzero $|\epsilon|$.

Answer 2

What you want to find is an addition that would make $\det (A+B)\ne 0$. Now since $\det A$ is a polynomial in the elements of $A$ you would have that there are arbitrarily small such additions.

Answer 3

If you mean the the least $k$ such that changing the values of $k$ elements of $A$ results in an invertible matrix, then $n-r$, where $A$ has order $n$ and rank $r$.