Timeline for How do you solve a tridiagonal matrix where all 3 diagonals are ones? [closed]
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 21, 2013 at 7:23 | comment | added | Carlo Beenakker | the formula for the inverse $M^{\rm inverse}$ of your $n\times n$ matrix $M$ is Eq. 10 of this 1996 paper phys.lsu.edu/~amarti9/adfaerf/… by Hu and O'Connell: $$M^{\rm inverse}_{ij}=-(-1)^{i+j}\frac{\cos[\frac{\pi}{3}(n+1-|j-i|)]-\cos[\frac{\pi}{3}(n+1-i-j)]}{2\sin\frac{\pi}{3}\sin[\frac{\pi}{3}(n+1)]}$$ this paper also gives more general expressions for the inverse when you add an arbitrary multiple of the unit matrix to your $M$. | |
| Jun 20, 2013 at 22:30 | history | closed |
Jack Huizenga Steven Landsburg Will Jagy Alexandre Eremenko Sergei Ivanov |
off topic | |
| Jun 20, 2013 at 21:41 | comment | added | Yemon Choi | The OP should clarify what he means by "solve" - does he want to invert this matrix? diagonalize it? | |
| Jun 20, 2013 at 21:24 | answer | added | Steven Landsburg | timeline score: 1 | |
| Jun 20, 2013 at 18:34 | comment | added | Carlo Beenakker | the formula for the inverse (for an arbitrary diagonal) is here: phys.lsu.edu/~amarti9/adfaerf/… | |
| Jun 20, 2013 at 18:33 | comment | added | Steven Landsburg | Noam: You give me far too much credit! | |
| Jun 20, 2013 at 18:26 | comment | added | Noam D. Elkies | Presumably "solve" = invert. This matrix is invertible iff its order is not $2\bmod 3$, in which case it has determinant $\pm 1$ and experiments such as (in gp) 1/matrix(19,19,i,j,abs(i-j)<2) suggest the inverse matrix always has $0,\pm1$ entries with tractable patterns. BTW the determinant depends on the order mod 6; perhaps that's what Steven Landsburg was hinting at? | |
| Jun 20, 2013 at 18:18 | comment | added | Steven Landsburg | Let's start with something simpler. How do you solve the number 6? | |
| Jun 20, 2013 at 18:04 | history | asked | Kenneth | CC BY-SA 3.0 |