Timeline for Result attribution for eigenvalues of a matrix of Pascal-type
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 1, 2017 at 4:43 | vote | accept | Alexander Burstein | ||
| Aug 11, 2017 at 17:50 | answer | added | esg | timeline score: 5 | |
| Aug 10, 2017 at 19:44 | comment | added | Alexander Burstein | @FedorPetrov My coauthor and I also looked at those eigenvectors for small $n$, but couldn't find a general formula for a given $k$ (except that $k=n$ is easy). Probably, some inclusion-exclusion going on there, but the exact formula is hard to guess. | |
| Aug 10, 2017 at 19:30 | comment | added | Fedor Petrov | The eigenvectors for $n$ at most 5 form a pretty nice orthogonal basis, but I do not see a general formula from them. | |
| Aug 10, 2017 at 19:23 | comment | added | Suvrit | @AlexanderBurstein btw the positive definiteness of the matrix $A_m$ in Sec. 4 of your paper can follow "more directly" by recalling that the Pascal matrix is positive definite (proof: $\binom{p+q}{p} = c\int_0^{2\pi} (1+e^{i\theta})^p(1+e^{-i\theta})^qd\theta$) and invoking Schur's theorem on psdness of the elementwise product of psd matrices. | |
| Aug 10, 2017 at 18:47 | history | edited | Alexander Burstein | CC BY-SA 3.0 |
improved formatting
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| Aug 10, 2017 at 18:34 | comment | added | Alexander Burstein | To see where this result might have been useful (and also, speaking of traces of matrices), take a look at Section 4 of this paper. @FedorPetrov | |
| Aug 10, 2017 at 18:32 | comment | added | Alexander Burstein | There is a paper by Amdeberhan that proves the determinant of this matrix is the product of those eigenvalues, but that does not seem to help in finding the individual eigenvalues. @FedorPetrov | |
| Aug 10, 2017 at 10:13 | comment | added | Fedor Petrov | By the way, computing the trace of this matrix and the sum of eigenvalues we get a famous identity $\sum \binom{2i}{i}\binom{2(n-i)}{n-i}=4^n$. | |
| Aug 10, 2017 at 8:04 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
more informative title
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| Aug 10, 2017 at 7:38 | history | edited | Alexander Burstein |
edited tags
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| Aug 10, 2017 at 7:36 | history | edited | Emil Jeřábek |
edited tags
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| Aug 10, 2017 at 7:34 | history | edited | Alexander Burstein |
added a tag
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| Aug 10, 2017 at 7:30 | review | First posts | |||
| Aug 10, 2017 at 7:37 | |||||
| Aug 10, 2017 at 7:26 | history | asked | Alexander Burstein | CC BY-SA 3.0 |