Timeline for Bounds for the difference in the number of ones in $M$ and $M^{-1}$
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Nov 25 at 19:55 | vote | accept | Simd | ||
| Nov 5 at 12:54 | history | edited | YCor | CC BY-SA 4.0 |
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| Nov 5 at 11:10 | answer | added | Fedor Petrov | timeline score: 8 | |
| Nov 5 at 1:10 | comment | added | Noam D. Elkies | One easy example: if $M$ has 1's on the diagonal and first off-diagonal (for a total of $2n-1$) then $M^{-1}$ is a triangular matrix with $(n^2+n)/2$ 1's. | |
| Nov 4 at 23:52 | answer | added | Bill Bradley | timeline score: 6 | |
| Nov 4 at 16:38 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 14:21 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 13:39 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 13:38 | comment | added | R.P. | So I see the following patterns: for $n=4,5,6,7$ we have that the maximum difference equals $(n-2)^2$, and this difference occurs between matrices with $2n+1$ and $(n-2)^2+2n+1=n^2-2n+5$ ones (but not necessarily exclusively). Moreover, the values $n+1,n+2,\ldots,2n-1$ are the first $n-1$ triangular numbers (starting from $0$). These assertions, if true for all $n>3$, look like they should be easily provable by exhibiting the appropriate matrices explicitly. This should be a way of gaining more insight into your question. | |
| Nov 4 at 13:32 | comment | added | Simd | @R.P. n=5 added. Let me know if any more examples would be useful. | |
| Nov 4 at 13:31 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 13:24 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (opinable...) + typo fixing (relation**s**hip instead of relationhip)
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| Nov 4 at 12:53 | comment | added | R.P. | So the maximum difference for $n=6$ is $16$ and the maximum for $n=7$ is $25$. Now I wonder if it will be $36$ for $n=8$... By the way do you have data for $n<6$ as well? | |
| Nov 4 at 12:17 | comment | added | Dietrich Burde | I would not say that it was asked "unsuccessfully". My answer there was for a question just asking whether the number of $1$'s in $A$ and $A^{-1}$ could differ. Later the question was edited. | |
| Nov 4 at 11:25 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 11:24 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing and typo fixing
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| Nov 4 at 10:53 | history | edited | Simd | CC BY-SA 4.0 |
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| Nov 4 at 10:46 | history | asked | Simd | CC BY-SA 4.0 |