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