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Hi all. I recently encountered a problem as following: if you are given a family of unitary matrices $A(t)$, in which $t\in\mathbb{R}$ is the parameter and especially $A(t)$ is smooth in $t$, which means the entries of $A(t)$ can be regarded as smooth functions of $t$.

So is there a way to label the eigenvalues of $A(t)$ as a series of functions $e^{i\theta_1(t)},\ldots, e^{i\theta_n(t)}$ such that they are all smooth in $t$, globally or at least locally?

I looked into Kato's perturbation of linear operator book but I didn't find this very problem. I though, on the other hand, think it is standard and should be true. So can anybody please hint me about any reference containing a possible proof to this result, if it is really true? Thanks in advance.

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  • $\begingroup$ 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. $\endgroup$
    – Alexander Chervov
    Commented May 8, 2012 at 8:04
  • $\begingroup$ Alexander, thanks a lot for your answer. Could you please be more specific about the paper of Arnold? THanks. $\endgroup$
    – Peng
    Commented May 8, 2012 at 17:58
  • $\begingroup$ 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. $\endgroup$
    – Dan Ramras
    Commented May 10, 2012 at 4:45
  • $\begingroup$ 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... $\endgroup$
    – Dan Ramras
    Commented May 10, 2012 at 4:48
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    $\begingroup$ 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. $\endgroup$
    – Peter Dalakov
    Commented May 15, 2012 at 17:48

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This is not an answer about the unitary case, but for an analytic family of selfadjoint matrices, the eigenvalues are analytic, according to Rellich's theorem, as described in the Linear Algebra book of P. Lax. Edit: but even if such a result holds for the unitary case, an analytic eigenvalue $\lambda(t)$ might not be expressible as $e^{i\theta(t)}$ because there are maps of $U(n)$ into $S^1$ which are not nullhomotopic.

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  • $\begingroup$ thank you guys, I will look into the literature you mentioned. Thanks a lot. $\endgroup$
    – Peng
    Commented May 16, 2012 at 6:26
  • $\begingroup$ This is tangentially related: springerlink.com/content/v24r1436w27n1115 $\endgroup$
    – Vít Tuček
    Commented May 16, 2012 at 12:20

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