Timeline for Eigencircles of n x n matrices?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2019 at 20:01 | answer | added | Bart Vanderbeke | timeline score: 0 | |
| S Jan 1, 2018 at 0:58 | history | suggested | jeq | CC BY-SA 3.0 |
Replaced quadruple backslashes with double backslashes, in an already-bumped question.
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| Jan 1, 2018 at 0:09 | review | Suggested edits | |||
| S Jan 1, 2018 at 0:58 | |||||
| Apr 27, 2012 at 5:57 | comment | added | john mangual | @Qiaochu Maybe these are 1D subspaces on which the action of the matrix is to rotate and dilate (i.e. multiply by $\lambda + \mu i$). A 2 x 2 matrix has exactly 2 complex eigenvalues and the pairs $(\lambda, \mu)$ can take a continuous range of values lying on a circle. So they have different cardinalities. | |
| Apr 27, 2012 at 3:45 | answer | added | Gottfried Helms | timeline score: 5 | |
| Apr 26, 2012 at 22:45 | comment | added | Qiaochu Yuan | @John: if $V$ is a 2D invariant subspace which contains no eigenvectors, then extending scalars to $\mathbb{C}$ you can find a pair of complex conjugate eigenvectors with complex eigenvalues which span $V \otimes \mathbb{C}$. In general just apply the structure theorem for f.g. modules over a PID to $\mathbb{R}[x]$. You get simple modules of the form $\mathbb{R}[x]/(x - r)$ (real eigenvectors) and $\mathbb{R}[x]/(x^2 + bx + c)$ for $b^2 - 4c < 0$ (pairs of complex eigenvectors). | |
| Apr 26, 2012 at 21:15 | answer | added | Federico Poloni | timeline score: 2 | |
| Apr 26, 2012 at 20:49 | comment | added | john mangual | @Qiaochu Does a real matrix have a $SO(2)$ worth of complex eigenvalues? These are 2D invariant subspaces, where the matrix acts as an element of $\mathbb{C} = SO(2) \times \mathbb{R}^+$. | |
| Apr 26, 2012 at 17:47 | comment | added | Victor Dods | Multiplication in the complex plane geometrically is rotation and scaling (by a positive constant). Maybe consider $SO(n) \times \mathbb{R}^+$, i.e. positively-scaled, special orthogonal matrices. Farr also mentions the use of quaternions toward the end of the article, though says this still applies to 2×2 matrices. Interesting stuff! | |
| Apr 26, 2012 at 16:14 | comment | added | Qiaochu Yuan | Isn't an "eigencircle" just a way to study complex eigenvectors of a real matrix without extending scalars? | |
| Apr 26, 2012 at 15:30 | history | asked | john mangual | CC BY-SA 3.0 |