Timeline for The height of the Perron-Frobenius eigenvector
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 24, 2014 at 18:35 | vote | accept | Seva | ||
| Mar 23, 2014 at 21:43 | answer | added | thomashennecke | timeline score: 5 | |
| Mar 23, 2014 at 12:27 | comment | added | Seva | @Federico: I could try to invent a similar diefinition of the height of a matrix, but it does not emerge naturally in the context I am interested in. The infinity norm as a height is just fine for my purposes, and other reasonable heights can be fine as well. | |
| Mar 23, 2014 at 11:13 | comment | added | Federico Poloni | probably you misunderstood me; I do know what an eigenvector is, I was asking you to clarify the definition of height, as you did (thanks!). I take it that there should be a similar definition of height or "smallness" for matrices then? What is it if it's not that matrix norm? | |
| Mar 23, 2014 at 9:33 | comment | added | Seva | @Federico: I do not define the notion of an eigenvector in my question, and the question is not about the norms of a matrix. The question is whether a "small" matrix has a "small" Perron-Frobenius eigenvector, under some reasonable definition of "smallness". As to the way I defined the height of a vector in ${\mathbb R}^n_+$: write the coordinates of the vector in a non-decreasing order, say $v_1,\ldots,v_n$, let $i$ be the largest index with $v^2_1+\dotsb+v^2_i<v^2_{i+1}+\dotsb+v^2_n$, and define then the height to be $v_n/v_{i+1}$. | |
| Mar 23, 2014 at 8:51 | comment | added | Federico Poloni | I am not familiar with the concept of height and I cannot make sense of your definition on the eigenvector. Could you please formalize it? For the height of the matrix, what you defined is called infinity-norm of a matrix, $\|A\|_{\infty}$, in matrix analysis (see for instance en.wikipedia.org/wiki/Matrix_norm). It satisfies some "typical" inequalities for a norm such as $\|Av\|\leq\|A\|\|v\|$, and one could probably find more results by looking it up with this name. | |
| Mar 23, 2014 at 6:50 | history | asked | Seva | CC BY-SA 3.0 |