Timeline for Minimum of squared sum minus sum of squares
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| May 15, 2015 at 19:12 | comment | added | Christian Remling | @squattyroo: To address your actual comment, if $A=D^2+\alpha xx^t$ with $\alpha\in\mathbb R$, then the calculation from the other answer now gives $G_{\alpha}=F/(1+\alpha F)$, and thus the eigenvalues of $A$ are now the solutions of $F(\lambda)=-1/\alpha$ (above we're in the special case $\alpha=-1$). The (interlacing) bounds stay the same though (if $\alpha>0$, there's no eigenvalue below the spectrum of $D^2$ and instead there's one above the spectrum). | |
| May 15, 2015 at 19:09 | comment | added | Christian Remling | @squattyroo: This (a rank-one perturbation with a coupling constant) is actually the situation I'm most familiar with. Somewhat unexpectedly, it's a pretty interesting subject; Barry Simon has a very readable review article on this, if you want to know more. | |
| May 15, 2015 at 17:31 | vote | accept | squattyroo | ||
| May 15, 2015 at 17:27 | comment | added | squattyroo | Very nice! Follow-up question: suppose we're looking at $D^2+\alpha xx^T$ for some $\alpha>0$. Then your analysis yields the largest eigenvalue is lower-bounded by $\alpha x_n^2$. It seems that it might be possible to analyze $F(\lambda)$ further and deduce a more interesting upper-bound than $\sum x_k^2$? | |
| May 15, 2015 at 1:35 | comment | added | Christian Remling | I should also point out that the inequality $x_{n-1}^2\le\lambda\le x_n^2$ is just the interlacing property, I don't need to look at resolvents for this. | |
| May 15, 2015 at 0:25 | history | edited | Christian Remling | CC BY-SA 3.0 |
added 2 characters in body
|
| May 15, 2015 at 0:18 | history | edited | Christian Remling | CC BY-SA 3.0 |
added 149 characters in body
|
| May 15, 2015 at 0:08 | history | undeleted | Christian Remling | ||
| May 15, 2015 at 0:07 | history | edited | Christian Remling | CC BY-SA 3.0 |
added 107 characters in body
|
| May 15, 2015 at 0:03 | history | deleted | Christian Remling | via Vote | |
| May 14, 2015 at 23:59 | history | answered | Christian Remling | CC BY-SA 3.0 |