Probability of invertibility of matrices
Robert Rosenbaum posted a really nice question to the UH math mailing list:
Given an
matrix of integers, what is the probability that it is
invertible?What if we put a bound on the magnitude of the integers (so that we consider only a finite number of matrices)? In other words, given an integer
and an integer
, what is the probability that an
of integers each less than
? I don’t know.
If we don’t restrict ourselves to integer valued matrices and consider all real or complex valued matrices, I think that the probability would be exactly 1. Can someone give an argument for this claim?
I got the last question, where you consider all of 
: from an earlier problem I posted about, it is clear that the set of noninvertible matrices has empty interior, so measure zero; therefore a random matrix is invertible with probability one.
The discrete case is probably a lot harder…
Update: It seems I made an unwarranted assumption. Just because the set of noninvertible matrices has an empty interior doesn’t mean it has measure zero. I still think that’s true, but the proof is not complete.
Possibly relevant posts:
- Solution to probability of matrix invertibility (5/18/2006)
- The measure of a manifold (5/24/2006)
- Invertibility of random matrices (12/23/2007)
The probability of a matrix being invertable is inversly proportional to the “square” of how useful its invertability would be.