Timeline for Parametrization of real diagonalizable matrices with given eigenvalues
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| May 6, 2014 at 13:34 | vote | accept | Ozz | ||
| May 5, 2014 at 9:37 | history | edited | Ozz | CC BY-SA 3.0 |
added 6 characters in body; edited title
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| May 5, 2014 at 8:07 | answer | added | Geoff Robinson | timeline score: 1 | |
| May 5, 2014 at 0:25 | comment | added | Nate Eldredge | @Geoff: Indeed, up to change of orthonormal basis, it suffices to take $\theta=\pi/2$. | |
| May 5, 2014 at 0:16 | comment | added | Geoff Robinson | I think it was known to Frobenius in the sense that he knew how to reduce to that form directly with real matrices without using complex matrices. If you use complex matrices, it's elementary linear algebra to see that it suffices to do the $2$-dimensional case, which is fairly easy to do directly. | |
| May 5, 2014 at 0:01 | comment | added | Geoff Robinson | Yes, I meant to indicate that whenever you have the complex diagonal form of the matrix, arranged so that complex conjugate eigenvalues succeed each other, you can replace each $2 \times 2$ diagonal sub-block corresponding to those eigenvalues with the $2 \times 2$ corresponding scaled rotation matrix. This works for diagonalizable matrices of any size. | |
| May 4, 2014 at 23:55 | comment | added | Ozz | Thanks, Geoff! Would you remember a reference for this? That matrix looks like a Givens rotation. Can I use Givens rotations for higher dimensional parametrizations? | |
| May 4, 2014 at 23:45 | comment | added | Geoff Robinson | Yes, there is, and it was known to Frobenius if my memory is correct. Replace $2 \times 2$ diagonal matrix with eigenvalues $re^{i\theta}$ and $r e^{-i \theta}$ by the matrix $\left( \begin{array}{clcr} r\cos \theta & r\sin \theta\\-r\sin\theta & r\cos \theta \end{array} \right). $ | |
| May 4, 2014 at 23:33 | history | asked | Ozz | CC BY-SA 3.0 |