Timeline for smooth dependency of eigenvalues of parametrized unitary matrice
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 16, 2012 at 2:51 | answer | added | Bob Terrell | timeline score: 1 | |
| May 15, 2012 at 17:48 | comment | added | Peter Dalakov | A tangentially related question was discussed in V.I.Arnold's paper "On matrices depending on parameters". There he gave a "normal form" (not Jordan normal form) into which you can put a holomorphically varying family of complex-valued matrices by a holomorphic change of parameters. | |
| May 10, 2012 at 4:48 | comment | added | Dan Ramras | I also recall looking in Kato's book way back when. I think I found there some examples showing that even for analytic paths of matrices, it's not always possible to find smooth paths of eigenvectors but I don't recall Kato addressing the question of eigenvalues. And I don't recall finding any other good references... | |
| May 10, 2012 at 4:45 | comment | added | Dan Ramras | Many years ago I wrote down a "proof" that one could find continuous functions of the desired form. Probably the functions I constructed were piecewise smooth in A(t) is smooth; and singularities would come at points where two distinct eigenvalues merge. I think I still have my notes on this, but I recall looking back at them and thinking there was a gap. Anyway, if piecewise-smooth would be of interest to you, I can try to dig up the argument. I think it just involved some fiddling with symmetric products in order to deal with merging eigenvalues. Nothing deep. | |
| May 8, 2012 at 17:58 | comment | added | Peng | Alexander, thanks a lot for your answer. Could you please be more specific about the paper of Arnold? THanks. | |
| May 8, 2012 at 8:04 | comment | added | Alexander Chervov | Locally no problem and no need of unitarity. Globally I almost sure that no. I am not sure, but Berry's phase is related to this for symmetric matrices - at least something like that is discussed in one of V.I. Arnold's papers. | |
| May 8, 2012 at 7:56 | history | asked | Peng | CC BY-SA 3.0 |