If $A(t)$ is an analytic, Hermitian matrixvalued 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 $[0,1] \times [0,1]$.
1 Answer
See assertions (L) and (M) of the main theorem of
 Andreas Kriegl, Peter W. Michor, Armin Rainer: DenjoyCarleman 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/s1185601400075. (pdf)

$\begingroup$ Thanks for the reference. These results are for unbounded operator valued functions. Are stronger results available for finitedimensional matrix valued functions? $\endgroup$ Aug 22, 2014 at 16:38

$\begingroup$ Finite dimensional results are essentially the same. $\endgroup$ Aug 22, 2014 at 17:23

$\begingroup$ 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