If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and eigenvectors of $A(t)$. My question is: what if $A$ is an analytic function of two or more real variables? Does anyone know any relevant references? More specifically, in my case $A = A(s,t)$ is analytic in a neighborhood of $[0,1] \times [0,1]$.
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1$\begingroup$ Eigenvalues are roots of characteristic polynomial, with coefficients that analytically depend on $(s,t).$ $\endgroup$– Dmitry SavostyanovAug 22, 2014 at 15:43
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See assertions (L) and (M) of the main theorem of
- Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011). (pdf)
Essentially it says, that you can have either:
real analytic eigenvalues and eigenvectors after finitely many blow ups and power substitutions of the parameters
eigenvalues or eigenvectors which are special functions of bounded variation, if you do not want to mess with the parameterizations.
For the $C^p$ case, see
- Differentiable roots, eigenvalues, and eigenvectors, Israel J. Math., doi:10.1007/s11856-014-0007-5. (pdf)
answered Aug 22, 2014 at 15:54
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$\begingroup$ Thanks for the reference. These results are for unbounded operator valued functions. Are stronger results available for finite-dimensional matrix valued functions? $\endgroup$ Aug 22, 2014 at 16:38
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$\begingroup$ Finite dimensional results are essentially the same. $\endgroup$ Aug 22, 2014 at 17:23
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$\begingroup$ This example was very illuminating: $$A(s,t) = \begin{pmatrix} s^2 & st \\ st & t^2 \end{pmatrix}.$$ Clearly A(s,t) is analytic, but the eigenvectors cannot be chosen to be analytic in a neighborhood of the origin. This is from the paper by Kurdyka and Paunescu in the references of the first paper you linked. Much appreciated. $\endgroup$ Aug 23, 2014 at 3:08