# How to correctly normalize modulus and phase of an eigenvector?

I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem: $$\lambda M x = J x.$$ I obtain complex eigenvalue and eigenvectors from Arpack using inverse shift method. I would like to compare the eigenvectors obtained with different meshes. To do this it is necessary to normalize norm and phase in a proper way. As concerns the norm of the eigenvector I normalize it imposing: $$x^HMx = 1.$$ Now, I would need an integral norm for the phase too. I tried one but it doesn't seem to work. I cannot obtain similar eigenvectors with two different meshes.

## 2 Answers

What do you mean by "compare eigenvectors"? If you just want to measure a concept of distance between them, I would use the angle between subspaces, which is independent of the normalization.

• I want to visualize the velocity field containted in the eigenvectors with Paraview. I would expect to see something similar or the same thing if the two eigenvectors are correctly normalized. Am I wrong? – Britomarti Jun 12 '15 at 18:02
• What kind of velocities do you have that have a complex phase? – Federico Poloni Jul 14 '15 at 17:57

Please refer to some work of Peter Arbenz: http://people.inf.ethz.ch/arbenz/publications.html

He had worked out some generalized eigenvalue problems with constraint condition (I remembered that he used the Jacobi–Davidson Algorithm), his model problems came from the FEM for Maxwell equations.

• There are 109 references in your link. Would you mind adding a summary of what you want to highlight from there? – nicoguaro Nov 22 '19 at 17:27