Timeline for Invertible matrices with bounded nonnegative coefficients
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2022 at 21:57 | comment | added | Andrea Marino | The link about 0,1 matrices is not that relevant to me, while the OEIS link for $2\times2$ matrices is nice. With a simple python program it's easy to see that the probability $p(n)$ of being non-singular with nonnegative coefficients up to $n$ is such that $p(n)/(1-1/n)$ is strictly decreasing and greater than $1$ (from some point on), so that one can conjecture that $p(n) \le (1+C)(1-1/n)$ for some positive constant $C<1/40$ (since for $n=31$ we get less than $1/40$. I'll look at the other links soon | |
| Jun 10, 2022 at 3:28 | comment | added | Gerry Myerson | OK. So, have you had a look at the links I posted, and at the further links and references given at those links? Did you find anything useful that way? | |
| Jun 9, 2022 at 22:23 | comment | added | Andrea Marino | Regarding the size of $m,n$: I'd like to have an estimate in both variables (like a taylor expansion in $1/m, 1/n$), but it's ok if we fix the size of the matrix and we let the bound for the coefficients go to infinity. | |
| Jun 9, 2022 at 22:05 | comment | added | Andrea Marino | Yes, the question is somewhat vague... It was meant to be lighthearted :) I don't need this for any application, so a variant of the estimate is ok too! | |
| Jun 8, 2022 at 23:40 | comment | added | Gerry Myerson | Related: mathoverflow.net/questions/20534/… and mathoverflow.net/questions/18636/… | |
| Jun 8, 2022 at 23:34 | comment | added | Gerry Myerson | oeis.org/A062801 tabulates "Number of $2\times2$ non-singular integer matrices with entries from $\{\,0,\dots,n\,\}$" up to $n=33$. The entry for $n=33$ is $1327606$. | |
| Jun 8, 2022 at 23:24 | comment | added | Gerry Myerson | oeis.org/A055165 is "Number of invertible $n\times n$ matrices with entries equal to $0$ or $1$." The tabulation there only goes up to $n=8$, for which the entry is $10160459763342013440$. There are some links and references there that may be useful. | |
| Jun 8, 2022 at 23:15 | comment | added | Gerry Myerson | The answer depends on $m$ and on the size of the matrix. Are we fixing one of these, and letting the other go to infinity? or are we letting both go to infinity in some precisely defined manner? or are we asking for a formula/estimate valid for all values of both variables? | |
| Jun 8, 2022 at 20:47 | comment | added | Sam Hopkins | The question is written in a confusing way, but it is asking for a lower bound on the number of matrices with entries in $\{0, \ldots, m-1\}$ that are nonsingular as real matrices (and points out that in the case of $m$ prime we get a bound by considering the finite field). | |
| Jun 8, 2022 at 20:10 | history | asked | Andrea Marino | CC BY-SA 4.0 |