Timeline for Anti-bidiagonal matrix with main anti-diagonal {1,2,3,...} and first sub-anti-diagonal {-1,-2,-3,...} has eigenvalues lambda={1,-2,3,-4,...}
Current License: CC BY-SA 3.0
7 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 28, 2014 at 1:01 | comment | added | Suvrit | @RichardZhang: I constructed that $V$ matrix, because in that question, the OP wanted all eigenvectors of the matrix; eigenvalues he already knew how to obtain; You may also want to have a look at page 3 of math.mit.edu/~plamen/files/KoevDopicoSR.pdf --- in particular, the relation of the two anti bi-diagonal matrices mentioned in there. | |
| Oct 28, 2014 at 1:01 | vote | accept | Richard Zhang | ||
| Oct 28, 2014 at 0:39 | comment | added | Richard Zhang | In my initial question, I was looking for a relatively short proof of the eigenvalue proposition, without necessarily constructing the eigenvectors. As you have shown, the eigenvectors are fairly complicated, thus leaving a direct $B_n x = \lambda x$ proof filling many pages. Instead, your $V$ matrix gets us there indirectly, since $VP^{-1}V^{-1}$ is a triangular matrix. To prove that it is triangular, we can do a proof by induction: project every $B_n$ onto the final two columns of their respective $V$. This turns out to be exactly what Darij had suggested in his comment. | |
| Oct 27, 2014 at 21:29 | comment | added | Suvrit | @RichardZhang: You can download the matlab file linked in my answer to get the closed form eigenvalues and eigenvectors of the matrix $P$ in my other answer; the eigenvectors for your matrix are then easily obtained I think as: $JV^TS^T$, for matrices $V$ and $S$ computed by my matlab code. | |
| Oct 27, 2014 at 21:26 | comment | added | Richard Zhang | I just saw your response. And wow it looks both right and extremely elegant. Nice. Let me quickly code this up to verify. | |
| Oct 27, 2014 at 21:24 | history | answered | Suvrit | CC BY-SA 3.0 |