Timeline for How to show a certain determinant is non-zero
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 28, 2014 at 16:28 | comment | added | darij grinberg | This proof can be made even simpler: If the determinant of the big matrix were $0$, then there would be a nontrivial $\mathbb{R}$-linear combination of its columns that would equal $0$. This immediately means that a nontrivial exponential polynomial with $n$ terms has at least $n$ distinct real roots; this completes the proof, with no induction needed. (Combined with the correct part of my answer, this yields that the determinant is positive.) | |
| Mar 27, 2013 at 22:16 | comment | added | Noam D. Elkies | Thanks. We also need that the coefficients of this exponential polynomial are not all zero; but each coefficient is an order $n-1$ determinant of the same type, so we can frame this as an argument by induction, and then at the $n-1$ step we showed that all of the coefficients are nonzero, so we're done. | |
| Mar 27, 2013 at 20:36 | history | answered | Todd Trimble | CC BY-SA 3.0 |