MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am having matrix $M_0$ with coresponding eigenvectors and 4 eigenvalues {0,0,a,-a}. Eigenvalue $\lambda=0$ is double degenerated. Now I am appliing small perturbation $\epsilon M_1$ and want to get the expansion of zero eigenvalues $\lambda$ like series of small parameter $\epsilon$.

My first thought was to apply eigenvalue deriviate theory and get corrections in a form: $\lambda(\epsilon)=\lambda_0+\epsilon\frac{\vec{w}M_1\vec{u}}{\vec u \vec w}$ (here u,w are left and right eigenvectors). However I have read that this method is not appropriate for eigenvalues having multiplivity higher than 1.

The question is how I can get series expansion for zero eigenvalues in my case?

UPD. Have looked Kato's book one more time and still can't get the idea how the eigenprojections are obtained. Resolvent of the the perturbed matrix $M(\epsilon)=M_0+\epsilon M_1$ is given by $R(\epsilon,\zeta)=(M(\epsilon)-\zeta)^{-1}$. Next step is obtaining the eigenprojection : $P=-\frac{1}{2 \pi i} \int{R(\epsilon,\zeta)d\zeta}$ where integration is carried along small circle around eigenvalue of unperturbed matrix $M_0$, $\lambda_0$. At this point I can't understand how to carry out this integration.

share|cite|improve this question
Have you checked Kato's book Perturbation theory of linear operators? – Liviu Nicolaescu Feb 7 '13 at 20:44
2… – Carlo Beenakker Feb 7 '13 at 22:59
I have cheked Kato's book but find it to difficult for me to understand. I have tried to apply quantum mechanics perturbation theory in degenerate case but the first step is failed: corrections to the degenerate eigenvalues in my case are complex, this have no sence. – Denys Feb 8 '13 at 6:15

Part (L) of the theorem in the following paper answers your question in the case that of normal matrices.

Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416. (pdf)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.