Timeline for Extremal properties of the determinant for matrices with entries in a fixed subset of $[-1,1]^{n^2}$?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2011 at 4:27 | answer | added | Will Orrick | timeline score: 2 | |
| Dec 8, 2011 at 9:41 | comment | added | Roland Bacher | Kevin: Thank you for these precisions. | |
| Dec 7, 2011 at 21:09 | comment | added | Kevin P. Costello | Roland: Sorry for being misleading by giving rationals...the maximum is only near that multiset, not exactly there (I had initially only searched in a mesh around that point, and didn't find any larger values). If you assume that (1) The maximum really occurs where there's five $1$s and a $−1$ and (2) that at the maximum the same two permutations have minimum determinant as in the $(0.25,−0.25,−0.64)$ case, you can solve explicitly for the maximum minimum determinant, getting that it occurs at $(2−\sqrt(5),\sqrt(5)−2,\frac{1−\sqrt{5}}{2})$ and has value $5 \sqrt{5} -11 \approx 0.1803$. | |
| Dec 7, 2011 at 15:52 | comment | added | Roland Bacher | @Kevin P. Costello: Are you sure that you get a local maximum for your multiset? It is obvious that local maxima consist of algebraic numbers but it would be somwhat surprising that there are rational solutions. | |
| Dec 6, 2011 at 22:52 | comment | added | GH from MO | @Tony: Thanks for the correction. I will delete my comment. | |
| Dec 6, 2011 at 22:38 | answer | added | Gerhard Paseman | timeline score: 0 | |
| Dec 6, 2011 at 11:13 | comment | added | Tony Huynh | By my previous comment we have the $\epsilon$-improvement that $a(n)$ is strictly less than $n^{n/2}$ for $n \geq 3$ since $det(A)=n^{n/2}$ if and only if $A$ is a Hadamard matrix. | |
| Dec 6, 2011 at 7:36 | comment | added | Tony Huynh | @GH: I don't think Hadamard's Conjecture is relevant here since we have to take the minimum determinant over all matrices with entries in $S$. Since each entry of a Hadamard matrix is 1 or -1, by rearranging the entries, we clearly get a zero determinant for $n \geq 4$, so this definitely won't be where the maximum occurs. | |
| Dec 6, 2011 at 7:17 | comment | added | Kevin P. Costello | I tried doing a computer search, and there seems to be a local maximum for the $n=3$ case near $\{1,1,1,1,1,1/4, -1/4, -16/25, -1\}$, where the minimum determinant has value $9/50$. I don't really see any compelling explanation for those values though. | |
| Dec 6, 2011 at 7:09 | history | edited | Roland Bacher | CC BY-SA 3.0 |
edited title
|
| Dec 6, 2011 at 6:59 | comment | added | Roland Bacher | Gerry Myerson, thank you. I have corrected this. | |
| Dec 6, 2011 at 6:58 | history | edited | Roland Bacher | CC BY-SA 3.0 |
deleted 5 characters in body
|
| Dec 5, 2011 at 22:39 | comment | added | Gerry Myerson | I think matrices have entries, not coefficients. | |
| Dec 5, 2011 at 22:23 | comment | added | Gerhard Paseman | Further, there was a sci.math post a few years back by Hugo Pfoertner on looking at matrices with entries in 1,2,...,n^2, and looking at variations of the determinant function when n=3. I think his intuition on this problem is worth something. Gerhard "Ask Me About System Design" Paseman, 2011.12.05 | |
| Dec 5, 2011 at 22:15 | comment | added | Gerhard Paseman | The math just rendered, and I see I have misread the question. I retract the above two comments, although you may find them useful. I conjecture a(n) near 0 for n larger than 2. Gerhard "These Are Not Binary Matrices" Paseman, 2011.12.05 | |
| Dec 5, 2011 at 21:42 | comment | added | Gerhard Paseman | And a(3) will be 4, a(4) will be 16, a(5) will be 48, and m(n) will be 0 or -a(n) when n is larger than 2, depending on whether you use absolute values or not. Gerhard "Ask Me About System Design" Paseman, 2011.12.05 | |
| Dec 5, 2011 at 21:37 | comment | added | Gerhard Paseman | This will likely be closed as a duplicate question. Will Orrick's maxdet site has info for n not divisible by 4. Gerhard "Ask Me About Binary Matrices" Paseman, 2011.12.05 | |
| Dec 5, 2011 at 18:42 | history | asked | Roland Bacher | CC BY-SA 3.0 |