Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?

Question

I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted answer there is a sum over all non-equal eigenvalues (the denominator is the difference of two eigenvalues, which would yield zero if the eigenvalues were equal).

1. Does that mean that if all the eigenvalues of a tensor are equal, the derivative of the eigenvector with respect to the components of the tensor could be taken as zero?
2. Or is it simply undefined? If so, could you please explain in simple terms the physical significance of having an undefined derivative of an eigenvector when all the eigenvalues are equal?

Thank you.

(Please note: This is my first time asking on this forum. Please let me know if I am using an incorrect format to ask questions and/or if I need to explain my question in more detail.)

Edit: I wanted to add a description of my particular use case. I am writing a subroutine to simulate a hyper-elastic material under certain loading conditions. The strain energy density of a "pure solid" is expressed in terms of the eigenvalues of the right Cauchy-Green deformation tensor. In order to calculate the stress of the material, I calculated the derivative of the strain energy density with respect to the Lagrangian strain tensor (which can be expressed in terms of the right Cauchy-Green deformation tensor), and I was able to form an equation for it correctly. After that, in order to calculate the constitutive material tensor, I need to calculate the derivative of the stress tensor with respect to the Lagrangian strain tensor again. Now, this process involves calculating the derivative of the eigenvectors of the right Cauchy-Green deformation tensor with respect to the tensor. Now, in case of distinct eigenvalues, the eexpression I get after differentiation seems to be correct. However, in my code; I still need to account for non-distinct eigenvalues and how the expression would change because of that.

Answer 1

I’ll try to explain it in physicists words:

A matrix with all eigenvalues equal is proportional to the identity matrix. The eigenvectors are maximally degenerate, as every arbitrarily oriented orthonormal base is eigenbase. If you now perturb one element of the matrix, you break the rotational symmetry, and the eigenbase “jumps” to the corresponding subspace. Therefore the derivative is undefined (“infinite”), just as the formulas in the cited thread suggest.