Let $\mathbb F$ be a field (ex. a finite field, or the reals), $A$ a $m\times n$ matrix over $\mathbb F$, and $x\in \mathbb F^n$ a vector. I'm interested in finding the smallest number of coordinates I need to change in $x$ to order to have $Ax=\mathbb 0$.

What would be an algorithm to find this, or alternatively, an intractability result (ex. NP-completeness)?

Note: I am thinking of this in the context of matrix rigidity.


This problem is NP-hard in general.

This is equivalent to finding the sparsest vector $y\in \mathbb F^n$ such that $$Ay = -Ax$$

As finding a sparsest solution vector for an underdetermined system of linear equations is NP-hard, so is your problem.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.