Orthonormal eigenvectors

[sina-mansour]

Question/Support Request

Hi,

I'm using Lapy to generate Laplace Beltrami Operators of a surface mesh.

In my experience, the eigenvectors generated by lapy.solver.Solver(mesh).eigs are neither orthogonal, nor unit norm vectors. I wanted to check if this is expected behavior? Given that the mesh structure is a symmetric matrix, should the eigenvectors be orthonormal?

Screenshots

Here's a heatmap of the dot product of pairs of eigenvectors for the first 20 eigenvectors:

And here's what other tools (like numpy.linalg.eigs) would generate. Note that for orthonormal eigenvectors, the dot products on the diagonal are all one (because of the unit norm) and off-diagonal values are all zero (because of orthogonality). I think the Lapy eigenvectors should have ideally been orthonormal too:

Environment

• LaPy Version: Latest (as of June 2023)
• OS: Ubuntu

[m-reuter]

Hi,
this is a generalised eigenvalue problem A x=lambda B x (see https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem ) and the eigenvectors will be B-orthonormal: x1^t B x2 = 0 ...