Timeline for Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2021 at 15:00 | vote | accept | Adam | ||
| Oct 25, 2021 at 19:36 | vote | accept | Adam | ||
| Oct 25, 2021 at 19:36 | |||||
| Oct 25, 2021 at 19:18 | review | Suggested edits | |||
| Oct 25, 2021 at 21:14 | |||||
| Oct 25, 2021 at 17:20 | comment | added | Adam | @ Christian Remling : Thank you very much. | |
| Oct 25, 2021 at 15:26 | comment | added | Christian Remling | @Adam: The matrix representation of a linear map is with respect to a basis. Now you can just say that you choose as your first basis vector the eigenvector of the large eigenvalue (and take the orthogonal direction as the second basis vector, so that you have an ONB and can still compute operator norms in the usual way). Alternatively (essentially the same argument really), replace $A$ by $RAR^{-1}$, with $R$ chosen as a rotation matrix that moves that eigenvector to $e_1$. Since $R$ is orthogonal, conjugation by $R$ does not change the operator norm. | |
| Oct 25, 2021 at 5:49 | vote | accept | Adam | ||
| Oct 25, 2021 at 18:40 | |||||
| Oct 24, 2021 at 21:22 | comment | added | Adam | @ Christian Remling That is interesting; I haven't seen it before. Could you please send a reference where I can see its proper proof? Thanks in advance | |
| Oct 24, 2021 at 19:38 | comment | added | Christian Remling | @Adam: This is the general matrix with $Ae_1=\lambda e_1$ and $\det A=1$. This form is more useful than $A=PDP^{-1}$ because it only has one additional parameter ($b$) rather than a full matrix ($P$). | |
| Oct 24, 2021 at 19:34 | comment | added | Adam | As far as I know, the matrix $A$ is conjugated to the diagonal matrix $D=diag(\lambda, 1/ \lambda)$, which means $A=PDP^{-1}$. I don't understand why you wrote $ A=\begin{pmatrix} \lambda & b \\ 0 & 1/\lambda \end{pmatrix} . $. | |
| Oct 24, 2021 at 19:20 | comment | added | Adam | Thank you very much for your answer. Could you please clarify the first paragraph of your answer ? Thanks in advance. | |
| Oct 24, 2021 at 18:05 | history | answered | Christian Remling | CC BY-SA 4.0 |