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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    $\begingroup$ Eigenvalues are roots of characteristic polynomial, with coefficients that analytically depend on $(s,t).$ $\endgroup$ Aug 22, 2014 at 15:43

1 Answer 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)
  • $\begingroup$ Thanks for the reference. These results are for unbounded operator valued functions. Are stronger results available for finite-dimensional matrix valued functions? $\endgroup$
    – Brian Lins
    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. $\endgroup$
    – Brian Lins
    Aug 23, 2014 at 3:08

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