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 $[0,1] \times [0,1]$. 



Eigendecomposition of analytic Hermitian matrix-valued functions of several variables
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)](http://www.mat.univie.ac.at/~michor/DC-perturb.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)](http://www.mat.univie.ac.at/~armin/publ/root-fin-publ.pdf)


